Sequential Equilibrium

Sequential equilibrium is David Kreps and Robert Wilson’s 1982 answer to a defect that Subgame Perfect Equilibrium cannot repair: when a game has no proper subgames, subgame perfection reduces to Nash equilibrium and stops refining anything. Their fix changes what an equilibrium is. An equilibrium is no longer a strategy profile; it is an assessment — a pair (μ, π) consisting of a strategy profile π and a system of beliefs μ that says, for each information set, how likely each node in it is given that the set has been reached. Two conditions are then imposed: sequential rationality, that every player’s continuation strategy is optimal at every information set given those beliefs, and consistency, that the beliefs are the limit of beliefs computed by Bayes’ rule from completely mixed strategies converging to π. Kreps and Wilson state the shift in one sentence: “The point to be stressed here is that an assessment (and not simply a strategy) will or will not be a sequential equilibrium. We require that an equilibrium specify beliefs as well as strategies” (Sequential Equilibria, Econometrica 50(4), 1982, pp. 863–894, fetched and read in full).

Mental Model

The whole concept follows from noticing one thing about probability. When a player finds herself at an information set that the equilibrium strategies reach with probability zero, Bayes’ rule offers her nothing. The conditional probability P(x | h) has the form P(x) / P(h) = 0/0. It is not that the answer is hard to compute; it is that no answer is defined. Yet the player has to move, and to move rationally she has to believe something about where she is.

So the theory must supply the beliefs from outside probability theory. That is the entire content of sequential equilibrium.

flowchart TB
    Q["a player is at information set h"]
    P{"is P_pi(h) > 0 ?"}
    B["Bayes' rule applies:<br/>mu(x) = P(x) / P(h)<br/>beliefs are FORCED"]
    Z["P_pi(h) = 0<br/>Bayes gives 0/0<br/>beliefs are UNDEFINED"]
    C["consistency: mu must be the limit<br/>of Bayes beliefs from completely<br/>mixed strategies converging to pi"]
    R["sequential rationality:<br/>given mu, the continuation<br/>strategy is optimal at h"]
    E["sequential equilibrium<br/>= consistent + sequentially rational"]
    Q --> P
    P -->|yes| B
    P -->|no| Z
    Z --> C
    B --> C
    C --> R --> E
    style Z fill:#ffecec
    style E fill:#e6f5e6

Figure 1 — the fork that generates the whole concept. The insight to take away: the left branch is uncontroversial and does no refining — everyone already agreed that on-path beliefs come from Bayes. All of the theory’s content, and all of its difficulty, is on the right branch, where the theory must invent a probability that the model does not determine. Consistency is the invention rule. Different inventions give different solution concepts, which is why perfect Bayesian equilibrium, sequential equilibrium, trembling-hand perfection, properness and stability are all different names for different rules on the right branch.

The second picture worth holding is the refinement ladder, but with the reason for each rung attached, because “sequential equilibrium is stronger than subgame perfection” is a fact that teaches nothing on its own.

flowchart TB
    N["Nash<br/>no profitable deviation<br/>evaluated at the root"]
    S["Subgame perfect (Selten 1965)<br/>Nash in every PROPER SUBGAME"]
    Q["Sequential (Kreps-Wilson 1982)<br/>optimal at every INFORMATION SET<br/>given consistent beliefs"]
    T["Trembling-hand perfect (Selten 1975)<br/>a best response along SOME sequence<br/>of completely mixed profiles"]
    N -->|"defect: certifies<br/>incredible threats"| S
    S -->|"defect: VACUOUS when there<br/>are no proper subgames"| Q
    Q -->|"defect: tolerates some<br/>weakly dominated play"| T
    NOTE["KW Prop 5: every perfect equilibrium is sequential.<br/>KW Thm 3: for generic payoffs they nearly coincide.<br/>KW Prop 6: sequential = 'weak perfect', exactly."]
    T -.- NOTE
    style Q fill:#ffe9cc

Figure 2 — the ladder, annotated with the failure each rung repairs. The insight to take away: the rungs are not a tower of increasing severity for its own sake; each exists because of a specific small game on which the rung below gives an answer nobody believes. The two upper rungs are far closer than the picture suggests — Kreps and Wilson prove that “for generic payoffs u every strict sequential equilibrium is perfect,” and that a mild reformulation of Selten’s definition makes the two sets identical. Subgame Perfect Equilibrium develops the lower rungs; this note owns the top two.

Why Beliefs Are Unavoidable: Selten’s Horse, Worked Exactly

Kreps and Wilson motivate the concept with a three-player game they take from Selten’s 1975 paper, universally known as Selten’s horse for the shape of its tree. The payoffs below are read from Gambit’s machine-readable transcription of Selten’s Figure 1 (catalog/journals/ijgt/selten1975/fig1.efg), whose header comment describes it as “the motivating example for his definition of (trembling-hand) perfect equilibrium, by showing a game that has an equilibrium which is ‘unreasonable’, but which is not ruled out by subgame perfection because this game has no proper subgames.”

flowchart LR
    R(("1")) -->|"A"| X(("2"))
    R -->|"D"| H2(("3"))
    X -->|"a"| T1["(1, 1, 1)"]
    X -->|"d"| H1(("3"))
    H1 -->|"l"| T2["(4, 4, 0)"]
    H1 -->|"r"| T3["(0, 0, 1)"]
    H2 -->|"l"| T4["(3, 2, 2)"]
    H2 -->|"r"| T5["(0, 0, 0)"]
    H1 -.->|"one information set:<br/>player 3 cannot tell<br/>these two nodes apart"| H2
    style H1 fill:#ffe9cc
    style H2 fill:#ffe9cc

Figure 3 — Selten’s horse. Payoffs are (player 1, player 2, player 3). Player 1 chooses A (across, to player 2) or D (down, to player 3). Player 2 chooses a (across, ending the game) or d (down, to player 3). Player 3’s single information set contains both of the shaded nodes. The insight to take away: the only node that begins a proper subgame is the root, because player 3’s information set straddles the tree. Subgame perfection therefore has exactly one subgame to check — the whole game — and so coincides with Nash equilibrium on this game. Any defect Nash has, subgame perfection here inherits.

The unreasonable equilibrium. Kreps and Wilson: “One Nash equilibrium for this game has player 1 choosing D, player 2 choosing a, and player 3 choosing l. This equilibrium is subgame perfect, as the only proper subgame here is the game itself. But, as Selten argues, this equilibrium is not sensible.”

Running that assessment through the checker built below:

=== the equilibrium Selten calls unreasonable: (D, a, l)
   Bayes beliefs at h3: {after A: 1, after D: 1}
   Nash?  root value 3
    ok at h1 (player 0): value 3
    NOT sequentially rational at h2 (player 1): 1 < 4 via ['d']
    ok at h3 (player 2): value 2

Read that middle line carefully, because it is the whole argument. Player 2’s information set is a singleton. It is reached with probability zero — player 1 plays D — but singleness means the belief is forced anyway: a distribution over one node puts probability 1 on it. So player 2’s conditional expected payoff is computable without inventing anything, and it says: playing a yields 1, while playing d yields 4 (because player 3, at the node d leads to, plays l, giving (4,4,0)). Player 2 is leaving 3 units on the table at a decision he might have to make. Kreps and Wilson: “Given the supposed behavior of 3, 2 prefers to choose d. (The reader can verify that if 1 realizes this, then 1 would optimally choose A instead of D, thereby upsetting the equilibrium.)”

Verify that last parenthesis: if player 2 plays d and player 3 plays l, then A gives player 1 a payoff of 4 against D’s 3. It does upset the equilibrium.

The sensible equilibrium, and what pins it down. Kreps and Wilson assert without derivation: “The only ‘sensible’ equilibrium in this game has 1 choosing A, 2 choosing a, and 3 choosing r with probability at least 3/4.”

Scanning the two free parameters — player 3’s belief on the node reached by d, and player 3’s probability of r — over a grid of twelfths:

scan: for which (belief on the d-node, prob of r) is (A,a,.) sequentially rational?
   feasible pairs: 8
   min belief on the d-node over all feasible pairs: 2/3
   at that minimum belief, the r-probabilities that work: [3/4, 5/6, 11/12, 1]
   for beliefs strictly above it, r-probabilities that work: [1]

That reproduces Kreps and Wilson’s 3/4 exactly and, more usefully, explains where it comes from — something the paper’s text does not do. Two constraints are operating at once:

• A belief constraint on player 3. At his information set, l yields 0 at the d-node and 2 at the D-node; r yields 1 and 0. Writing β for the belief on the d-node, r is at least as good as l exactly when β·1 ≥ (1−β)·2, that is β ≥ 2/3. Below that threshold player 3 strictly prefers l, and no amount of mixing helps.
• A mixing constraint imposed by player 2. Player 2 must prefer a (worth 1) to d, which is worth 4·Pr[l]. So Pr[l] ≤ 1/4, i.e. Pr[r] ≥ 3/4.

The two bite in different places. For β > 2/3 player 3 strictly prefers r, so Pr[r] = 1 and player 2’s constraint is slack. The 3/4 figure is live only at β = 2/3 exactly, where player 3 is indifferent and free to mix. That is what the scan’s last two lines say.

This resolves an uncertainty flag left by a sibling note

Subgame Perfect Equilibrium carries a > [!warning] Uncertain callout saying the payoffs in Kreps and Wilson’s Figure 3 “OCR as an ambiguous block” and that its author therefore built a substitute example. The payoffs are now recovered from Gambit’s transcription of Selten (1975) Figure 1 and independently confirmed against four separate quantitative claims in the Kreps–Wilson text: that (D, a, l) is a Nash equilibrium; that player 2 prefers d given l; that player 1 then prefers A; and that the sensible equilibrium needs r with probability at least 3/4. All four check out. That callout can be retired.

Sequential Rationality, Symbol by Symbol

Kreps and Wilson build the definition in two steps, and the intermediate step is instructive because it almost works.

Step one: the singleton criterion. Since a singleton information set has forced beliefs, one can already demand: “A strategy π should be such that for any information set h that is a singleton, player t(h) should not be able to change his strategy unilaterally and thereby improve this expected utility starting from h.” That is exactly the test that killed (D, a, l) above. It requires no new machinery at all.

Why it is not enough. Their Figure 4 is the counterexample. Player 1 chooses L, R or A; A ends the game, while L and R both lead into a two-node information set where player 2 chooses l or r. Reconstructing the figure’s payoffs from the scan and checking them against the text’s three claims — l beats r at both nodes; A beats L and R when 2 plays r; L beats A and R when 2 plays l — gives A → (0,2), (L,l) → (2,1), (L,r) → (−1,−1), (R,l) → (1,1), (R,r) → (−2,0).

Their diagnosis: “A Nash equilibrium for this game has 1 choosing A, and 2 choosing r. This strategy is subgame perfect, and it satisfies the further criterion given above. But if 1 gives the move to 2, then regardless of what 2 thinks the chances are that he is at one node or the other in his information set, 2 will do better by choosing l.” Player 2’s action l weakly dominates r node by node — it pays 1 at both nodes, versus −1 and 0. No belief whatsoever can rationalise r. And yet the singleton criterion is silent, because player 2’s information set is not a singleton.

That is the exact gap the definition must close. Kreps and Wilson’s statement of the fix invokes Savage: “if 2 is rational in the sense of Savage — when faced with a choice 2 makes some assessment about what 1 did that is consistent with what 2 knows, and then optimizes accordingly — then 2 will choose l. This is the substance of sequential rationality: the strategy of each player starting from each information set must be optimal starting from there according to some assessment over the nodes in the information set and the strategies of everyone else.”

The formal objects. A system of beliefs is a function

   μ : X → [0,1]     with     Σ_{x ∈ h} μ(x) = 1   for every information set h ∈ H

Walking the symbols: X is the set of decision nodes; H is the set of information sets; h is one of them; μ(x) is “the probability assigned by player t(h) to x ∈ h if h is reached.” The constraint is per information set, not global: μ is a collection of conditional distributions, one for each h, and it says nothing about the relative likelihood of two different information sets.

An assessment is the pair (μ, π), where π is a behaviour strategy profile. Given an assessment, the conditional distribution over terminal nodes given that h was reached is defined as you would expect: P^{μ,π}(z | h) = 0 if z is not a terminal successor of h; and otherwise, if p^n(z) ∈ h (that is, z is n moves past a node of h),

   P^{μ,π}(z | h) = μ(p^n(z)) · Π_{m=1..n} π( a( p^{m-1}(z) ) )

Reading it: start with the belief that you are at the particular node of h that leads to z, then multiply by the probability of each action along the path from that node to z. The p^m(·) notation is Kreps and Wilson’s m-th predecessor. Nothing subtle happens here — this is just “weight by the belief, then play forward.”

Sequential rationality, finally:

   E^{μ,π}[ u_{t(h)}(z) | h ]  ≥  E^{μ,π'}[ u_{t(h)}(z) | h ]
        for all h ∈ H and all π' with π'_j = π_j for j ≠ t(h)

“In words, taking the beliefs as fixed, no player prefers at any point to change his part of the strategy π.” Two things about the quantifier that are easy to misread. The deviation π' may change the player’s behaviour at every information set he owns, not just at h — this is a full continuation-strategy comparison, not a one-shot deviation test. And the condition is required at every h ∈ H, including the information sets of players who will never reach them, which is the whole point.

Consistency, and the Ladder of Weaker Notions

Sequential rationality alone is not enough, because with beliefs unconstrained you could justify almost anything. Consistency is the constraint. Kreps and Wilson’s definition is short and, on first reading, unmotivated:

Let Π⁰ be the set of all strictly positive strategies. That is, π ∈ Π⁰ if π(a) > 0 for all a ∈ A. If π ∈ Π⁰, then P^π(x) > 0 for all x, and the only reasonable way to define beliefs μ associated with π is via Bayes’ rule: μ(x) = P^π(x) / P^π(H(x)). Let Ψ⁰ denote that subset of the set of assessments (μ, π) where π ∈ Π⁰ and μ is defined from π by Bayes’ rule.

Definition. An assessment (μ, π) is consistent if (μ, π) = lim_{n→∞} (μ_n, π_n) for some sequence {(μ_n, π_n)} ⊂ Ψ⁰. … A sequential equilibrium is an assessment (μ, π) that is both consistent and sequentially rational.

Unpacking: Π⁰ is the set of behaviour strategies that play every action with strictly positive probability. Under such a strategy every node is reached with positive probability, so Bayes’ rule is always defined and beliefs are forced. Ψ⁰ is the set of assessments built this way. Consistency says: your assessment must be a limit point of that set. The strategies may go to zero on some actions in the limit, but the beliefs must be the limits of the beliefs computed before they got there.

Kreps and Wilson concede immediately that “this definition of consistency is not completely intuitive on its own. We propose it because it neatly embodies a number of distinct intuitive notions of consistency.” Their §5 then walks up a ladder of those notions. Knowing the ladder is worth more than memorising the definition, because applied papers use the weaker rungs and call the result “perfect Bayesian equilibrium.”

flowchart TB
    C1["(5.1) Bayes where possible<br/>mu(x) P(H(x)) = P(x)<br/>the minimum anyone demands"]
    C2["(5.2) Structural consistency<br/>SOME strategy pi' reaching h<br/>generates mu at h by Bayes"]
    C3["Lexicographic consistency<br/>a shared, h-independent sequence of<br/>fallback hypotheses pi(2), pi(3), ..."]
    C4["(5.3) Bayes extended across<br/>one player's own information sets<br/>a first defection does not make a second likelier"]
    C5["(5.4) the same, across players<br/>rational beliefs are common knowledge<br/>implies Bayes inside proper subgames"]
    C6["FULL CONSISTENCY<br/>limit of completely mixed assessments<br/>no correlation in defections<br/>+ common knowledge of secondary hypotheses"]
    C1 --> C2 --> C3 --> C4 --> C5 --> C6
    W["KW's own warning:<br/>'By dropping (5.4), subgame perfection is lost.'"]
    C5 -.- W
    style C6 fill:#ffe9cc

Figure 4 — the consistency ladder. Each rung implies all the rungs below. The insight to take away: “perfect Bayesian equilibrium,” as used in most applied work, sits somewhere on the lower rungs — usually (5.1) plus whatever the author needs — and the choice is not innocuous. Kreps and Wilson flag the sharpest consequence in a parenthesis: drop (5.4) and you lose subgame perfection, so the “refinement” would no longer refine the concept it was meant to strengthen.

The two examples that force the upper rungs are worth stating, because they are the reason the definition is a limit and not something simpler.

Beliefs must cohere across one player’s own information sets. In their Figure 6, player 2 has two information sets, one leading to the other, and holds beliefs at the second that are not what he would compute by carrying his own first-set beliefs forward through the strategies. “If player 2 reaches his first information set and adopts the beliefs shown, then he expects (given 3’s strategy) to reach his second information set. And if he uses Bayes’ rule starting from his first information set together with 3’s strategy in order to obtain his beliefs in the second, he would not come up with the beliefs shown.” The remedy is condition (5.3), whose philosophy they state as: “A first defection does not make a second more likely; correlation in defections are (partially) ruled out.”

Beliefs must not attribute unnecessary defections. Their Figure 7 shows player 3 reaching an unexpected information set and explaining it by supposing that both player 1 and player 2 changed strategy, when supposing that only player 2 defected would suffice. “If one grants the principle that defections from the equilibrium strategy ought to be uncorrelated — that given that 2 has defected (which he surely must have, given that 3’s information set has been reached), the most likely (in a lexicographic sense) hypothesis is that 1 continues to play according to π — then the answer is no.” They sharpen it with a thought experiment worth remembering: “This example becomes even more stark if we suppose that players 2 and 3 are the same. Then the defection by 2 is known to 3 — should this player revise his assessments as to what 1 did because he himself defected?”

Full consistency handles both, by requiring convergence “to the strategy π everywhere, including those parts of the tree that could be reached only if an initial defection occurred.”

It is worth recording that the authors were not comfortable with their own definition. Their closing remark on §5: “Upon studying the first part of the Appendix, the reader may well conclude that we have required too much consistency in beliefs off the equilibrium path. … We shall proceed here to develop the properties of sequential equilibrium as defined above; however, we do so with some doubts of our own concerning what ‘ought’ to be the definition of a consistent assessment.” Forty-plus years on, that hesitation looks prescient; see the section on the refinement programme below.

Perfect Bayesian Equilibrium: Why “Bayes Whenever Possible” Is Not a Definition

The rung of the ladder that applied work actually stands on deserves its own treatment, because “perfect Bayesian equilibrium” is not one concept and the phrase that usually defines it does not survive contact with a general game tree. The cleanest primary treatment is Julio González-Díaz and Miguel Meléndez-Jiménez’s On the Notion of Perfect Bayesian Equilibrium (TOP 22, 2014, pp. 128–143; DOI 10.1007/s11750-011-0239-z), which sets out to fix exactly this. Their framing: “Often, perfect Bayesian equilibrium is loosely defined by stating that players should be sequentially rational given some beliefs in which Bayes rule is applied ‘whenever possible’. We argue that there are situations in which it is not clear what ‘whenever possible’ means.”

Start at the bottom rung, which does have an agreed definition:

Definition 1 (weak perfect Bayesian equilibrium). An assessment (b, μ) is a weak perfect Bayesian equilibrium if it is sequentially rational and, on the path of b, μ is derived from b by Bayes rule.

Nothing at all is required off the path. The authors note the provenance — “This equilibrium concept was introduced by Myerson (1991) when preparing the ground for the definition of sequential equilibrium”, where “originally, Myerson called this equilibrium concept weak sequential equilibrium” — and the damage: “although it is an equilibrium concept stronger than Nash equilibrium, a weak perfect Bayesian equilibrium does not even need to be subgame perfect.” This is the same casualty Kreps and Wilson name when they write “by dropping (5.4), subgame perfection is lost.”

Why “whenever possible” is not well defined. Their Example 1 is the whole argument. Player 1 chooses U or D at w₁ and plays D in equilibrium. If he had played U, player 2 would move at w_{2,1}, playing d. Player 3 has an information set w₃ containing three nodes: x₁ (reached via U then u), x₂ (via U then d), and x₃ (via D). The set w₃ is off the path, so weak perfect Bayesianism says nothing. Does “whenever possible” say anything?

• Reading A: no. w₃ has probability zero under b, and “we cannot condition on probability 0 events.”
• Reading B: yes, partly. “Given b and conditional on either x₁ or x₂ being reached, i.e., conditional on player 1 having played U, any belief consistent with Bayes rule should put probability 0 at x₁, since player 2 is playing d at w_{2,1}.”

And even under Reading B — which the authors call “the natural interpretation” — “Bayes rule imposes no restriction on the relative probabilities of x₂ and x₃.” The phrase therefore neither determines a belief nor even determines whether it is meant to apply. That is why several inequivalent “perfect Bayesian equilibria” coexist in the literature: Harris and Townsend (1981) in mechanism design, Fudenberg and Tirole (1991) for multistage games with observed actions, and Battigalli’s (1996) strategic independence family. Of Fudenberg and Tirole specifically, González-Díaz and Meléndez-Jiménez are precise about both the scope and the strength: “in order to decide what restrictions on beliefs are reasonable, they use the specific structure of the games in the class to which they have restricted. Somehow, by formally disentangling the meaning of ‘whenever possible’ in multistage games with observed actions and independent types, they get to something that is very close to sequential equilibrium.” Very close — but defined only on that class, and not extendable to general extensive games. Anyone who writes “PBE” without saying which PBE has left the reader to guess among these.

Their repair, and where it sits. Rather than formalise “whenever possible,” they generalise the subgame idea. Given an information set w, the quasi-subtree Γ_w is everything that comes after w. It is regular if no information set straddles its boundary — “if an information set v has a node that comes after w, then all the nodes in v come after w.” A regular w therefore defines a genuine game G_w(μ) in which Nature first picks a node x ∈ w with probability μ(x). When w is a singleton this is exactly a proper subgame.

Definition 2 (simple perfect Bayesian equilibrium). An assessment (b, μ) is a simple perfect Bayesian equilibrium if, for each regular information set w, the restriction of (b, μ) to G_w(μ) is a weak perfect Bayesian equilibrium.

Their own one-line summary: “simple perfect Bayesian equilibrium is to weak perfect Bayesian equilibrium in imperfect information games what subgame perfection is to Nash equilibrium in perfect information games.” And their Proposition 1 places it exactly:

Claim	Statement
(i)	Every sequential equilibrium is a simple perfect Bayesian equilibrium
(ii)	Every simple perfect Bayesian equilibrium is a weak perfect Bayesian equilibrium
(iii)	Every simple perfect Bayesian equilibrium is subgame perfect
(iv)	Every extensive game with perfect recall has at least one simple perfect Bayesian equilibrium

The proofs are short and instructive. (i) holds because Kreps–Wilson consistency “requires that the beliefs μ are the limit of the beliefs associated with a sequence of completely mixed behavior strategies converging to b. Clearly, this implies that Bayes rule must be applied on the path of all games associated with regular information sets.” (ii) holds because the root is itself a regular information set and G_root(μ) = G. (iii) holds because “there is a one to one correspondence between subgames and games defined from regular information sets that are singletons”, and simple PBE demands a weak PBE there where subgame perfection demands only a Nash equilibrium. (iv) is a corollary of (i) plus Kreps and Wilson’s existence theorem — which is worth noticing: the weaker concept inherits its existence proof from the stronger one.

flowchart TB
    NE["Nash equilibrium"]
    SPE["Subgame perfect equilibrium<br/>(Nash in every proper subgame)"]
    WPBE["Weak PBE (Myerson's<br/>'weak sequential equilibrium')<br/>Bayes ON the path only"]
    SPBE["Simple PBE<br/>weak PBE in every game induced by a<br/>REGULAR information set"]
    FT["Fudenberg-Tirole PBE<br/>multistage games with observed<br/>actions only — 'very close to<br/>sequential equilibrium'"]
    SEQ["SEQUENTIAL EQUILIBRIUM<br/>consistency = limit of completely<br/>mixed Bayes beliefs"]
    PERF["Trembling-hand perfect"]
    SEQ --> SPBE
    SPBE --> WPBE
    SPBE --> SPE
    SPE --> NE
    WPBE --> NE
    PERF --> SEQ
    FT -.->|"defined only on<br/>a restricted class"| SEQ
    style SEQ fill:#ffe9cc
    style WPBE fill:#f2f2f2

Figure 4b — where “perfect Bayesian equilibrium” actually sits. An arrow A → B reads “every A is a B”. The insight to take away: weak perfect Bayesian equilibrium and subgame perfect equilibrium are incomparable — neither implies the other — which is why the loose “PBE” of applied papers can fail to be a refinement of subgame perfection at all. Simple PBE is the cheapest concept that dominates both, and sequential equilibrium sits strictly above it. Note also what the diagram cannot show: Fudenberg and Tirole’s PBE is not a rung on this ladder for general games, because it is only defined for multistage games with observed actions.

One honest reason to stop short of sequential equilibrium. It is not always laziness. González-Díaz and Meléndez-Jiménez point out a real limitation: “there are natural economic settings in which it cannot even be defined. The notion of consistent beliefs cannot be (trivially) extended to games in which the players have a continuum of strategies; the models of auctions being the most outstanding example of the latter type of games.” Consistency is a statement about limits of completely mixed strategies, and “completely mixed” over a continuum is not the same object. This is why auction theory — see Bayesian Games and Types — works with Bayes–Nash equilibrium and belief restrictions rather than with consistency.

Consistency in Action: When It Bites and When It Does Not

Abstract definitions of consistency are easy to nod along to and hard to apply. Two small computations settle what it actually does.

Case one: consistency pins the belief exactly. Kreps and Wilson’s Figure 5 shows a fragment where Nature moves first, choosing left with probability 1/3 and right with probability 2/3. Player 1 cannot tell the two nodes apart — they are in one information set — and at that set he plays C with probability 1 and D with probability 0. Both D branches lead into player 2’s information set, which therefore has prior probability zero.

Their claim: “player 2’s beliefs in his information set must attach probability 1/3 to the left-hand node. This is because player 1 cannot distinguish between the two nodes in his information set, so any strategy he could hypothesize that gives 2’s information set positive probability must (by Bayes’ rule) preserve the initial probability assessment.”

Verified by explicit trembles, in exact arithmetic:

KW Figure 5: player 1 trembles onto D with probability eps
   eps = 1/10:      beliefs at player 2's set = 1/3, 2/3
   eps = 1/1000:    beliefs at player 2's set = 1/3, 2/3
   eps = 1/1000000: beliefs at player 2's set = 1/3, 2/3

The belief is 1/3 for every ε, not merely in the limit, because player 1 has one tremble parameter and it cancels: (1/3)ε / ((1/3)ε + (2/3)ε) = 1/3. This is what Kreps and Wilson mean by “simply assuming that the players’ beliefs always respect the informational structure of the game constrains players’ beliefs.” An off-path belief that put 1/2 on the left node would not be consistent, and no story about trembles could rescue it.

Case two: consistency leaves the belief completely free. Return to Selten’s horse. Player 3’s information set is off the path when player 1 plays A and player 2 plays a. Reaching it requires a defection, but two different players can supply one, and consistency places no restriction on their relative rates.

Selten's horse, same trembling exercise (player 1 trembles at eps, player 2 at eps^k):
   player 2 trembles at eps^1: belief on the d-node -> 999/1999      (0.499750)
   player 2 trembles at eps^2: belief on the d-node -> 999/1000999   (0.000998)
   player 2 trembles at eps^3: belief on the d-node -> 999/1000000999 (0.000001)

   a tremble sequence producing EXACTLY the critical belief 2/3:
      eps = 1/10:      belief on the d-node = 2/3
      eps = 1/1000:    belief on the d-node = 2/3
      eps = 1/1000000: belief on the d-node = 2/3
   one driving it to 1 (player 2 trembles at a strictly slower rate):
      eps = 1/10:   belief on the d-node = 99/109
      eps = 1/1000: belief on the d-node = 999999/1000999

Any value in [0,1] is a consistent belief here, obtained by choosing how fast player 2’s tremble vanishes relative to player 1’s. The sequence ε₁ = ε, ε₂ = 2ε/(1−ε) lands exactly on the critical 2/3 for every ε, and ε₁ = ε², ε₂ = ε drives it to 1.

The lesson, and it is the important one: consistency is a real constraint when the zero-probability event has a single explanation, and no constraint at all when it has several. In Figure 5 only player 1 could have deviated, and Bayes’ rule inside his information set does the rest. In Selten’s horse either player 1 or player 2 could have deviated, and the theory has nothing to say about which. Consistency does not select the sensible equilibrium in Selten’s horse — it merely permits the beliefs that support it.

An Assessment Checker

The definition is mechanical enough to implement, and implementing it is the fastest way to stop confusing the three conditions. The checker below takes a game, a behaviour profile and a belief system, and verifies sequential rationality at every information set. All arithmetic is fractions.Fraction, for the same reason as in Bayesian Games and Types: sequential rationality turns on indifference, and floating point cannot see indifference.

from fractions import Fraction as F
from itertools import product
 
class Game:
    """Nodes are dicts.  Terminal: {'pay': (u0,u1,...)}.  Chance: {'chance': {act: prob},
    'kids': {act: node}}.  Decision: {'pl': i, 'iset': name, 'kids': {act: node}}."""
    def __init__(self, root, nplayers):
        self.root, self.nplayers = root, nplayers
        self.isets = {}                 # iset name -> dict(player, actions, members[path])
        self._index(root, ())
 
    def _index(self, nd, path):
        if 'pay' in nd: return
        if 'chance' in nd:
            for a, k in nd['kids'].items(): self._index(k, path+(('N',a),))
            return
        h = nd['iset']
        rec = self.isets.setdefault(h, {'pl': nd['pl'], 'actions': list(nd['kids']),
                                        'members': []})
        assert rec['pl'] == nd['pl'] and rec['actions'] == list(nd['kids']), \
            f"information set {h} is malformed"
        rec['members'].append(path)
        for a, k in nd['kids'].items(): self._index(k, path+((h,a),))
 
    def realiz(self, pi):
        "P^pi(x) for every node, keyed by the path that reaches it."
        out = {}
        def walk(nd, path, p):
            out[path] = p
            if 'pay' in nd: return
            if 'chance' in nd:
                for a, k in nd['kids'].items():
                    walk(k, path+(('N',a),), p*nd['chance'][a])
            else:
                for a, k in nd['kids'].items():
                    walk(k, path+((nd['iset'],a),), p*pi[(nd['iset'],a)])
        walk(self.root, (), F(1))
        return out
 
    def bayes_beliefs(self, pi):
        """Beliefs from Bayes' rule where defined; None where Bayes is silent."""
        R = self.realiz(pi); mu = {}
        for h, rec in self.isets.items():
            tot = sum(R[m] for m in rec['members'])
            for m in rec['members']:
                if tot != 0:
                    mu[m] = R[m]/tot          # Bayes' rule applies
                elif len(rec['members']) == 1:
                    mu[m] = F(1)              # forced: a singleton set sums to 1 anyway
                else:
                    mu[m] = None              # Bayes is SILENT here -- must be supplied
        return mu
 
    def cond_payoff(self, h, pi, mu, player):
        "E^{mu,pi}[ u_player | h ], per Kreps & Wilson section 4."
        rec = self.isets[h]; tot = F(0)
        for m in rec['members']:
            if mu[m] == 0: continue
            tot += mu[m] * self._exp_from(self.at(m), pi, player)
        return tot
 
    def seq_rational(self, pi, mu, verbose=False):
        ok = True
        for h, rec in self.isets.items():
            i = rec['pl']
            here = self.cond_payoff(h, pi, mu, i)
            best, arg = here, None
            for dev in self.pure_profiles(i):        # every pure plan for player i
                alt = dict(pi); alt.update(dev)
                v = self.cond_payoff(h, alt, mu, i)
                if v > best: best, arg = v, dev
            if best > here:
                ok = False
                if verbose:
                    used = {k for k, v in arg.items() if v == 1}
                    print(f"    NOT sequentially rational at {h} (player {i}): "
                          f"{here} < {best} via {sorted(a for _, a in used)}")
        return ok

Commentary on the parts that encode a definition rather than a convenience:

• bayes_beliefs returns None, not zero, where Bayes is silent. This is the single most important line in the file. A solver that silently substitutes a uniform belief at unreached information sets is not implementing sequential equilibrium; it is implementing one arbitrary refinement of it. Returning None makes the caller confront the choice, and it makes the TypeError on None * Fraction a feature: forget to supply an off-path belief and the program stops.
• Singleton information sets get belief 1 even off the path. The constraint Σ_{x∈h} μ(x) = 1 forces it, and this is exactly what makes Kreps and Wilson’s intermediate singleton criterion work without any consistency apparatus. Getting this wrong is why a first implementation reports Selten’s horse as fine.
• seq_rational compares against every pure plan for the deviating player, not against single action changes. That is what the definition’s quantifier says. The one-shot deviation shortcut used for Subgame Perfect Equilibrium is a theorem there, not a definition, and it is not available here without proof.
• cond_payoff skips members with belief zero before recursing. Not an optimisation: it avoids evaluating a subtree that contributes nothing, which matters when a belief of zero sits over an infinite-payoff branch in a modified game.
• The assert in _index catches the most common data-entry bug in extensive games: two nodes assigned to the same information set with different action sets or different owners. Such a game is not well-formed, and without the check the solver produces plausible-looking nonsense.

The checker does not verify consistency. That is deliberate and worth saying explicitly: consistency is a statement about the existence of a sequence, and no finite computation decides it in general. What the checker gives you is realiz and bayes_beliefs, which are enough to test a candidate sequence, which is what the tremble computations above do. Deciding consistency exactly requires algebraic machinery — the standard route is Kohlberg and Mertens’ characterisation via the structure of the limit set, and in practice people verify it by exhibiting a tremble sequence, as done here.

Properties

Kreps and Wilson’s §6 establishes the results that make the concept usable. Each matters for a different reason.

Result	Statement	Why it matters
Proposition 1	“For every extensive game, there exists at least one sequential equilibrium.”	A refinement that can empty the solution set is useless. This is the licence to use it. It follows from Selten’s 1975 existence theorem for perfect equilibria plus Proposition 5.
Proposition 2	The correspondence from initial assessments and payoffs to sequential equilibria is upper hemi-continuous.	Small perturbations of the data cannot make an equilibrium appear from nowhere. This is what makes comparative statics legitimate.
Proposition 3	“If (μ, π) is a sequential equilibrium, then π is a subgame perfect Nash equilibrium.”	Sequential equilibrium really does sit above subgame perfection on the ladder. Note the direction: it is the strategy that is subgame perfect, the beliefs having done their work already.
Proposition 4	Sequential equilibria are extended subgame perfect — Nash in every subform, not merely every proper subgame, with the initial distributions on subform roots related to each other by Bayes.	This is the technically correct statement of “it keeps refining where subgame perfection runs out.” The converse is false, and their Figure 8 gives a game where every Nash equilibrium is extended subgame perfect but the depicted one “could never be part of a sequential equilibrium: no matter what beliefs player 2 has in his information set, B is a better action than A.”
Theorem 1 / generic structure	For generic payoffs, the set of sequential equilibria is finite and almost every one is strict.	The pathologies — continua of equilibria, isolated points — are non-generic. Their Figures 9 and 11 exhibit both, so “generic” is doing real work.

The failed converse in Proposition 4 is the one to remember, because it is a compact statement of the concept’s added value: a game can pass every subgame-flavoured test and still contain an action that no belief could ever justify.

Trembling-Hand Perfection, and How It Differs

Selten’s 1975 trembling-hand perfect equilibrium attacks the same problem by a different route, and the relationship is closer than the ladder picture suggests.

Kreps and Wilson restate Selten’s definition in their own vocabulary. Say that a convergent sequence {(μ_n, π_n)} ⊂ Ψ⁰ justifies the consistent assessment (μ, π) that is its limit. Then:

An assessment (μ, π) ∈ Ψ is a perfect equilibrium if it is justified by a sequence {(μ_n, π_n)} for which, for each player i and each index n, π_i is an optimal response for player i to the other players’ strategies (π_j^n)_{j≠i}.

The difference from sequential equilibrium is one clause, and Kreps and Wilson name it precisely: “One can interpret this definition as the composition of two criteria. The first … is that each player i’s strategy is an optimal response to the assessment (μ, π) and that, moreover, (μ, π) is a fully consistent sequential equilibrium. The second requires that for some sequence that justifies (μ, π), each player’s strategy is a robust best response. … It is the second of these two criteria that we forego in the definition of a sequential equilibrium.”

Robustness against the trembles themselves is what kills weakly dominated strategies. Their own comparison to Selten: “Selten’s definition accomplishes two things at once: It implicitly generates beliefs at information sets off the equilibrium path, and it requires that players’ strategies be optimal with respect to those beliefs. In addition, it eliminates from consideration strategies that are otherwise weakly dominated. In a sequential equilibrium, the former is explicitly done, and the latter is dropped.”

A minimal example where they differ. Kreps and Wilson note a corollary that makes the gap easy to exhibit: “In a normal form game, every Nash equilibrium is sequential.” So take any 2×2 game with a Nash equilibrium in weakly dominated strategies, draw it in extensive form with a two-node information set for the second mover, and you have one.

	L	R
T	(1, 1)	(0, 0)
B	(0, 0)	(0, 0)

(B,R): player 2's information set is reached with probability 1, so Bayes applies:
   beliefs: {T-node: 0, B-node: 1}
   sequentially rational at every information set: True
   -> (B,R) with these beliefs is a SEQUENTIAL EQUILIBRIUM.

Is it trembling-hand perfect?
   opponent plays L with prob 1/2:       u(T) = 1/2,       u(B) = 0, B best response: False
   opponent plays L with prob 1/10:      u(T) = 1/10,      u(B) = 0, B best response: False
   opponent plays L with prob 1/1000000: u(T) = 1/1000000, u(B) = 0, B best response: False

(B, R) is a sequential equilibrium: player 2’s information set is on the path, beliefs are forced, and both players are indifferent, so nobody has a strict improvement. It is not trembling-hand perfect, because against any strictly positive mixture by player 2, T earns a positive amount and B earns zero — so B is a best response to no completely mixed profile, and no justifying sequence exists. The diagnosis is weak dominance: T weakly dominates B, with a strict gain of 1 against L and a tie against R.

How much daylight is there really? Very little, and Kreps and Wilson prove it three ways.

• Proposition 5: every perfect equilibrium is sequential. The inclusion is strict.
• Theorem 3: “For any fixed extensive form and initial assessment, for generic payoffs u every strict sequential equilibrium is perfect.” Combined with their Theorem 1 — that generically almost every sequential equilibrium is strict — this paraphrases to “for almost every game, almost every sequential equilibrium is perfect.” The exceptions are confined to weak equilibria and only off the equilibrium path, and the sets of equilibrium paths coincide exactly.
• Proposition 6: if you weaken Selten’s definition to allow vanishing uncertainty about the payoffs rather than only about the actions — a weak perfect equilibrium, where “π_j need only be a best response to the perturbed strategies for some payoffs for j that are ‘close’ to u” — then “the sets of weak perfect and sequential equilibria coincide,” exactly.

So the practical difference is not which equilibria you get. It is which is easier to work with, and Kreps and Wilson are candid that this was their main motive: “The first is pragmatic: In many examples of interest … it is vastly easier to verify that a given equilibrium is sequential than that it is perfect.”

A third notion hiding in the definitions: perfection of the tree versus perfection of the matrix. Kohlberg and Mertens state the convention that makes “perfect equilibrium of a tree” meaningful: “a perfect (proper) equilibrium of a tree is a perfect (proper) equilibrium of its agent normal form” — the normal form in which each information set is treated as a separate player, so a tremble at one information set is independent of a tremble at another. Perfection of the ordinary (reduced) normal form is a different requirement, and the two do not coincide. Selten’s own counterexample is Figure 2 of the 1975 paper, and it can be read exactly rather than squinted at, because the Gambit project keeps it as machine-readable test data (catalog/journals/ijgt/selten1975/fig2.efg, fetched 2026-08-29). The file’s own header comment states what it is for: “This is a counterexample presented in Sel75, to show that extensive and normal form concepts of perfectness do not coincide. This game has one perfect equilibrium in the extensive form, but a distinct (pure) strategy equilibrium is also perfect in the normal form.”

The tree, transcribed from the .efg node list (which is depth-first, with children in the order the action list gives them):

   I --R--> (1, 1)
   |
   L
   |
   II --R--> (0, 2)
   |
   L
   |
   I --r--> (0, 3)
   |
   l
   |
       (2, 0)

   Perfect information: every information set is a singleton.
   Player I owns two of them; player II owns one.

Backward induction, in exact arithmetic: at player I’s second information set l pays 2 and r pays 0, so I plays l; player II therefore compares R (which pays him 2) against L (which leads to l, paying him 0), so II plays R; at the root I compares R (pays 1) against L (pays 0), so I plays R. The outcome is (1, 1).

Now reduce to the normal form. Player I has three reduced strategies — R, Lr, Ll — and player II has two:

	II plays R	II plays L
R	(1, 1)	(1, 1)
Lr	(0, 2)	(0, 3)
Ll	(0, 2)	(2, 0)

An exhaustive scan of this 3×2 matrix gives exactly one pure Nash equilibrium, (R, R). But R as a normal-form strategy is silent about what I would do at his second information set — it merely says the game ends at the root — so it is compatible with the behaviour r there, which the agent normal form forbids outright (l strictly dominates r for that agent). That is the non-coincidence: the extensive form pins down behaviour at an unreached information set; the reduced normal form does not, and its perfection concept therefore admits strategies whose tree-level realisation is not perfect. Player II’s side shows the same slack from the other direction: against R he is exactly indifferent, so the equilibrium component is {R} × {any mixture putting weight ≥ 1/2 on R}, the 1/2 being the point where I becomes indifferent between R (worth 1) and Ll (worth 2 with probability 1 − p). Gambit’s documented outputs sit precisely on and inside that set: gambit-lcp returns the boundary point NE,1,0,1/2,1/2,1/2,1/2 in behaviour strategies, and gambit-liap returns NE,0.998701,0.000229,0.001070,0.618833,0.381167 in the reduced strategic form, with p ≈ 0.619 (Gambit CLI documentation, read 2026-08-29).

Uncertain

Verify: that Selten’s Figure 2 in the original paper carries these payoffs, and the exact statement of the extensive-versus-normal-form claim in his own words. Reason: Selten (1975), IJGT 4, pp. 25–55 could not be retrieved. https://link.springer.com/content/pdf/10.1007/BF01766400.pdf returns HTTP 200 with Content-Type: text/html — a redirect to link.springer.com/article/...?error=cookies_not_supported, i.e. a challenge page, not the PDF. (This is the “check the extracted text, not the status code” failure in its purest form.) The tree and payoffs above are transcribed from Gambit’s .efg encoding, which is a faithful machine-readable transcription maintained by the project but is still a secondary rendering of Selten’s figure; the header comment quoted above is Gambit’s characterisation, not Selten’s. The backward induction, the Nash scan, and the p ≥ 1/2 threshold were computed here from that encoding and are independent of it only in the sense that they check for internal consistency. To resolve: obtain the IJGT original (a library copy, or the Springer chapter reprint in Models of Strategic Rationality, 10.1007/978-94-015-7774-8_1) and compare Figure 2.

Where the Refinement Programme Ran Out of Road

The second motive was the more consequential one, and it is also where the story ends badly. Making beliefs explicit was supposed to let analysts argue about which beliefs are plausible, and thereby narrow the equilibrium set further: “making explicit the construction of beliefs off the equilibrium path enables discussion of which beliefs are ‘plausible’ and which are not. Such discussion is difficult in the context of Selten’s mechanical and indirect procedure for generating beliefs.”

Their §8 is the demonstration, using an example due to Elon Kohlberg. Player 1 chooses A, L or R; A ends the game; L and R lead into a two-node information set where player 2 chooses l or r. Payoffs recovered from the scan: A → (1, 1), (L,l) → (2, −1), (L,r) → (−10, −2), (R,l) → (−1, −2), (R,r) → (0, −1).

Every quantitative claim Kreps and Wilson make about this figure checks out against that reconstruction, which is why it can be stated with confidence despite the OCR:

1) Is R strictly dominated by A for player 1?
   against l: A gives 1, R gives -1  ->  A strictly better: True
   against r: A gives 1, R gives  0  ->  A strictly better: True

2) equilibrium (L, l): beliefs forced by Bayes
   mu(x) = 1, mu(x') = 0        sequentially rational: True

3) equilibrium (A, r): the information set is off the path; scan the beliefs
   mu(x) values for which (A, r) is sequentially rational: 0, 1/12, 1/6, 1/4, 1/3, 5/12, 1/2

4) is that belief CONSISTENT?  Trembles of player 1 alone generate it:
   L nine times rarer than R: mu(x) -> 1/10
   L nine times likelier:     mu(x) -> 9/10

Item 3 reproduces Kreps and Wilson’s “supported by beliefs on the part of 2 that μ(x) < 1/2” and sharpens it: the exact threshold is μ(x) ≤ 1/2, the boundary included because player 2 is then indifferent. Item 4 is the punchline. Consistency does not rule this equilibrium out. Every belief in [0,1] is a limit of Bayes beliefs under completely mixed trembles by player 1 alone, so (A, r) supported by μ(x) = 1/10 is a bona fide sequential equilibrium.

And yet it is absurd. Kreps and Wilson: “Player 2 ‘ought’ not to conclude, upon reaching h, that 1 chose R, because R is dominated by A for 1. The only beliefs by 2 that make sense assign μ(x) = 1 (or, at least, > 1/2), which leads 2 to prefer l. Player 1, realizing this, prefers L.” The reasoning is forward induction, not backward: player 2 should infer from the fact that player 1 deviated at all that he deviated in the only way that could possibly pay.

They then show the argument survives a modification designed to break it. Replace A by a simultaneous bimatrix subgame whose unique Nash equilibrium pays player 1 exactly 1. Now A no longer guarantees more than R. “But playing A yields an expected equilibrium payoff of 1, which is more than 1 gets from R, so 2 ‘ought not’ to attach too much weight to the possibility that 1 played R given that 1 has not played A.”

This is the seed of the entire post-1982 refinement literature — Cho and Kreps’ intuitive criterion, divinity, universal divinity, Kohlberg and Mertens’ strategic stability, Myerson’s properness (which Kreps and Wilson already gesture at: “Myerson argues (essentially) that assessments off the equilibrium path should be such that the preponderance of weight goes to the ‘least costly’ mistakes”). And it is where the authors themselves put the brakes on:

We will not propose any formal criteria for “plausible beliefs” here. In certain cases, such as Myerson’s concept of properness, some formalization is possible. In other cases it is not clear that any formal criteria can be devised — it may be that arguments must be tailored to the particular game.

That sentence, in the founding paper, is where the road runs out. The refinement programme produced a long sequence of concepts, each motivated by a game on which the previous one gave a bad answer, and no consensus terminus. The Stanford Encyclopedia’s treatment of the same territory is instructive by omission: it defines sequential equilibrium loosely as beliefs “consistent with Bayes’s rule” and works its examples on Selten’s horse, without attempting to adjudicate the plausibility question at all (SEP, Game Theory).

Uncertain

Verify: the SEP entry’s definition of sequential equilibrium. Reason: it is weaker than Kreps and Wilson’s. SEP writes “a profile of strategies σ and a system of beliefs μ consistent with Bayes’ rule” — which is condition (5.1) on the ladder above, the bottom rung, and is what most textbooks call perfect Bayesian equilibrium. Kreps and Wilson’s consistency is the limit condition, strictly stronger, and the gap is exactly what Figures 6 and 7 of their paper exist to demonstrate. This note follows Kreps and Wilson. If you have read the SEP entry and formed the impression that consistency just means “use Bayes where you can,” that impression is wrong, and the Selten’s-horse computation above shows why the difference is not academic.

Invariance: the Same Game, Drawn Differently

Four years after Kreps and Wilson, Elon Kohlberg — the source of the example above — and Jean-François Mertens turned the argument into a structural objection with a name. Their On the Strategic Stability of Equilibria (Econometrica 54(5), 1986, pp. 1003–1037) lists “three conditions which seem necessary for strategic stability — backwards induction, iterated dominance, and invariance”. Sequential equilibrium satisfies the first and fails the other two.

Their Figure 2 is the game Γ(x), with x ≤ 2. Player I chooses T, M or B. T ends the game at (2,2). M and B lead into a two-node information set for player II, who chooses L or R:

                    L -> (3, 3)
             M --> [ node m ]
            /             R -> (x, 0)
   I ------ T ----------------> (2, 2)
            \
             B --> [ node b ]
                    L -> (0, 0)
                    R -> (1, 1)

   Γ(x),  x ≤ 2.  Dotted line (nodes m, b) = player II's single information set.

The reconstruction is confirmed by two independent numbers in the running text. Kohlberg and Mertens say II prefers R “if he assesses the conditional probability of the bottom point to be at least 3/4” — and indeed R beats L for II exactly when μ(b)·1 ≥ (1 − μ(b))·3, i.e. μ(b) ≥ 3/4. They also say B “is strictly dominated by T (and also by M for x > 1)” — B pays I either 0 or 1, T pays 2, and M pays 3 or x.

(T, R) is a sequential equilibrium. Kohlberg and Mertens verify consistency explicitly and by hand: “let (1−10ε, ε, 9ε) and (ε, 1−ε) be completely mixed strategies for players I and II, respectively. They converge to T for player I and R for player II, and the conditional probability on the bottom point of the information set converges to .9.” Nine-tenths on b clears the 3/4 threshold, so R is sequentially rational, and (T, R) survives.

It should not. Their forward-induction argument is the one Kreps and Wilson had already made informally: “player II knows that I will never choose B, which is strictly dominated by T … so if II sees he has to play, he should deduce that I … certainly did not choose B, where he was sure to get less than 2; player II should thus infer that I had in fact played M, betting on a chance to get more than 2 … and so player II should play L, and hence player I should play M.”

What is new in 1986 is the diagnosis. Redraw Γ(x) by coalescing moves — split player I’s single three-way choice into “stay or leave”, then a second information set where he picks between M and B. Nothing about the game changes: it has the same normal form. But now M strictly dominates B at a decision node, so sequential rationality itself forces I to choose M there, II’s belief is pinned to μ(m) = 1, and (T, R) is gone. Kohlberg and Mertens draw the conclusion in one sentence:

This example highlights a basic flaw in the concept of “sequential equilibrium”: it depends on all the arbitrary details with which the tree was drawn.

The formal backing is Thompson’s and Dalkey’s theorem that any two trees with the same normal form are connected by a chain of inessential transformations — coalescing of moves, inflation–deflation, addition of superfluous moves, interchange of simultaneous moves, and (with chance) replacing a nature move that leads only to terminal nodes by its expected payoff. “So any solution that is independent of those 5 categories of irrelevant details of the tree must depend only on the normal form. In particular, then, a strategically stable equilibrium of a game tree must be sequential in any other game tree having the same normal form.”

That requirement is not vacuous, because of their Proposition 0: “A proper equilibrium of a normal form is sequential in any tree with that normal form.” So Myerson’s properness — a normal-form notion — buys tree-independent sequentiality for free. It still is not enough: “the bad equilibrium 2,2 is proper in Γ(0)”, and the reader is walked one step further, to a tree in which player I is given an explicit coin-toss move between T and M, where “the only sequential equilibrium remaining is the good equilibrium 3,3.” The escalation ends with the paper’s set-valued concept, and with the summary judgement that “all the known solution concepts that satisfy backwards induction — sequential, perfect, and proper equilibrium — fail to satisfy the iterated dominance requirement.”

flowchart LR
    G1["Γ(x) as drawn:<br/>I picks T / M / B at one node"]
    G2["Γ(x) coalesced:<br/>I picks stay/leave,<br/>then M vs B at a second node"]
    G3["Γ(x) with an explicit<br/>coin-toss move between T and M"]
    S1["(T,R) IS a sequential equilibrium<br/>μ(b)=0.9 is consistent"]
    S2["(T,R) is NOT sequential:<br/>M strictly dominates B at a node,<br/>so μ(m)=1 is forced"]
    S3["(T,R) is NOT sequential:<br/>the coin toss strictly dominates B"]
    N["same normal form throughout"]
    G1 --> S1
    G2 --> S2
    G3 --> S3
    G1 -.-> N
    G2 -.-> N
    G3 -.-> N
    style S1 fill:#ffecec
    style S2 fill:#eefaee
    style S3 fill:#eefaee

Figure 6a — the invariance failure. The insight to take away: all three trees are the same game in the only sense that should matter — they have the same normal form and are related by transformations no player could detect — yet sequential equilibrium answers differently in each. The concept is a property of the drawing, not only of the game. That is a much harder objection than “it admits an implausible equilibrium,” and it is the reason strategic stability is set-valued and normal-form-based rather than another rung on the same ladder.

Two things follow for practice. First, if you report “the sequential equilibria of this game,” you are implicitly reporting a property of the extensive form you chose to write down; a colleague who coalesces two of your moves may get a different answer, and neither of you is wrong. Second, the fix that is cheap and generally worth taking is Proposition 0: computing a proper equilibrium of the normal form gives you an assessment that is sequential in every tree with that normal form. See Nash Equilibrium for the normal-form machinery this leans on.

The Deeper Problem: the Epistemic Foundation Does Not Hold Up

The Kohlberg example says the refinement is incomplete — it admits equilibria nobody believes. A sharper result says something worse about the whole family: the epistemic assumptions that make sequential rationality meaningful are, in most games, logically inconsistent.

Philip Reny’s Common Belief and the Theory of Games with Perfect Information (Journal of Economic Theory 59, 1993, pp. 257–274) puts the claim in the abstract: “The statement ‘it is common belief that player i is Bayesian rational’ (where Bayesian rational means an expected utility maximizer) is defined for two-person games with perfect information. It is shown that in most such games it is not possible for a theory to postulate the Bayesian rationality of all players and to be common belief. This bears directly upon the salience of standard solution concepts such as subgame perfect, and sequential equilibria.”

The two assumptions in question:

(a) each player believes that his opponent is an expected utility maximizer, so long as this is consistent with the history of play; (b) both players believe (a), both players believe that the other believes (a), etc.

Reny’s result: “no theory of finite two-person games with perfect information, unless it applies only to a rather small subset of such games, can contain both of the following assumptions without logical contradiction.” And the conclusion he draws, naming the concepts explicitly: “Since (as we shall argue) one of (a) or (b) must be absent from any theory of games, thereby potentially allowing non-backward induction outcomes, the plausibility of standard equilibrium concepts such as subgame perfection (Selten), perfection (Selten), sequential equilibrium (Kreps and Wilson), and others is called into question.”

flowchart TB
    A["(a) each player believes the other<br/>maximises expected utility,<br/>so long as the history allows it"]
    B["(b) (a) is common belief"]
    T["a theory of games covering<br/>most finite two-person<br/>perfect-information games"]
    X["LOGICAL CONTRADICTION<br/>Reny 1993: (a) and (b) cannot<br/>both hold in most such games"]
    D1["drop (a):<br/>a player may believe his opponent<br/>is irrational even when every past<br/>move was consistent with rationality"]
    D2["drop (b):<br/>rationality is believed but not<br/>COMMONLY believed"]
    R["either way, non-backward-induction<br/>play can be rational, and the<br/>refinements' motivating intuition weakens"]
    T --> A
    T --> B
    A --> X
    B --> X
    X --> D1 --> R
    X --> D2 --> R
    style X fill:#ffecec

Figure 6 — Reny’s impossibility, and the two exits. The insight to take away: this is not the usual “the refinement selects the wrong equilibrium” complaint. It is that the reasoning used to justify off-path optimality — “he would still be rational there, and I would still believe it, and he would know I believe it” — cannot be maintained in most perfect-information games, because reaching certain nodes is itself evidence against it. Reny’s own worked case is the game he calls TOL(n) (take-it-or-leave-it, a centipede), where for n > 3 “the necessary common beliefs are precluded by the game’s structure.”

The mechanism is a self-referential trap, and Reny spells it out on a four-move example: “At y, the only way player 2 can believe that player 1 is Bayesian rational, is if 2 believes that 1 believes that 2 will play D at y with high enough weight. … But if 2 is Bayesian rational, 2 will play D with probability zero. Thus, if at y, 2 believes that 1 is Bayesian rational, then 2 must also believe that 1 believes that 2 is not Bayesian rational.” The belief that sustains your opponent’s rationality is a belief in your own irrationality.

Reny also records why this is not merely academic, citing the reputation result: Kreps, Milgrom, Roberts and Wilson “show that if there is even the slightest possibility that one of the players is not an expected utility maximizer, then rational play need not consist solely of defecting at each stage” in the finitely repeated prisoner’s dilemma — “and generally, so long as expected utility maximization is not common knowledge, and the game is long enough, rational play need not coincide with backward induction … or even any game with perfect information and no indifference among terminal nodes.” The empirical failure of backward induction and the logical trouble with common belief are the same phenomenon. See Common Knowledge and Rationality Assumptions and Backward Induction, which develop the epistemic critique for the perfect-information case; the point here is that it does not stop at backward induction — it reaches the belief-based refinements too, because they are built on the same assumption in a more explicit form.

Resolved 2026-08-29

An earlier revision of this note flagged Reny’s Definition 2 and the exact scope of “most” as unverified, because dklevine.com/archive/refs4386.pdf is an image-only scan — pdftotext returns 18 characters and pdffonts lists no embedded fonts — and only part of it had been rendered. The remaining pages have now been rendered with pdftoppm -png -r 135 and read; the definitions and the theorem are transcribed below. The distinction that mattered was retrieval versus extraction: the file downloads fine and is completely legible as an image; only the text layer is missing.

Definition 2, and the machinery under it. Reny builds the impossibility from a node-by-node iteration rather than from a global epistemic model, which is why it is checkable on a small tree. Fix a two-person finite extensive game of perfect information. X is the set of decision nodes, X_i player i’s nodes, M_i player i’s set of mixed strategies. A strategy m_i reaches x if, for some m_{-i}, the profile (m_i, m_{-i}) gives the path through x positive probability. A conjecture c_i(x) ∈ M_{-i} is i’s belief, held at node x, about the strategy his opponent is using — held at every node, not only at his own.

Definition 1. (m_i, c_i) ∈ M_i × C_i satisfies Bayesian rationality if (i) ∀x ∈ X_i, c_i(x) reaches x; (ii) ∀x ∈ X_i, ∀x' ∈ X such that x immediately precedes x', and m_i reaches x', c_i(x) = c_i(x'); (iii) ∀x ∈ X_i reachable by m_i, m_i is a best response at x given c_i(x).

Reny’s own gloss: (i) “simply asks that a player’s conjecture be consistent with the information he possesses”; (ii) is “a particularly weak form of Bayesian updating” — your beliefs about the opponent do not change merely because you moved; (iii) “asks for expected utility maximizing behavior at all decision nodes not precluded by i’s strategy.”

The iteration then defines R_i^x(n), the set of i’s Bayesian-rational strategies reaching x that survive n rounds of “i believes −i is Bayesian rational at x, i believes that −i believes that i is …”. R_i^x(1) is just the Bayesian-rational strategies reaching x, and

R_i^x(n+1) = { m_i ∈ M_i | for some conjecture profile c_i,
                 (i)   m_i reaches x,
                 (ii)  (m_i, c_i) satisfies Bayesian rationality,
                 (iii) c_i(x) ∈ convex-hull( R_{-i}^x(n) ) }

The convex hull in (iii) is not decoration: Reny takes it from Pearce’s observation that for the purpose of an expected-utility calculation a belief over the opponent’s mixed strategies can be replaced by a single mixed strategy in the convex hull of its support, which is what lets c_i(x) live in M_{-i} rather than in a space of measures over M_{-i}.

Definition 2. A node x ∈ X is consistent with the common belief of Bayesian rationality (CBR) if ⋂_{n=1}^{∞} R_i^x(n) ≠ ∅ for i = 1, 2.

So “the node survives every round of the iteration” is the definition. Reny gives an equivalent fixed-point form he finds easier to use: a jointly rational belief system (JRBS) for x is a pair of nonempty sets (R₁*, R₂*), R_i* ⊆ M_i, such that m_i ∈ R_i* exactly when some conjecture profile c_i makes m_i reach x, makes (m_i, c_i) Bayesian rational, and puts c_i(x) in the convex hull of R*_{-i}. His Proposition: “A node x ∈ X is consistent with CBR if and only if there is a jointly rational belief system for x.”

Worked on the centipede. TOL(3) — take-it-or-leave-it, three moves, Figure 3 of the paper: player 1 at x takes (d₁, payoffs (1,0)) or passes (r₁) to player 2 at y, who takes (D, (0,2)) or passes (R) to player 1 at z, who takes (d₂, (3,0)) or passes (r₂, (0,3)). Reny walks the iteration: R₁^y(1) is the set of 1’s mixed strategies that give positive weight to r₁ at x and all weight to d₂ at z; both D and R are best responses for 2 to something in M₁, so R₂^y(1) = M₂; but by the recursion R₂^y(2) = {D}, “since 2’s conjecture at y about 1’s future play must be d₂, and D is 2’s unique best response at y given this. But this renders R₁^y(3) empty since 1’s conjecture at y must have support in R₂^y(2) and so must be D. But reaching y — which every strategy in R₁^y(3) must do — is then inconsistent with 1’s Bayesian rationality.” Three rounds, and the node is gone.

The theorem that quantifies “most.” Reny weakens (a) to (a′) — “if effective, each player, so long as it is consistent with the history of play, believes that his opponent is Bayesian rational” — where “ineffective” means the player “has a strategy that is best for him no matter how his opponent behaves”, i.e. a strictly dominant choice at that node. Call a node relevant if it is consistent with both players’ Bayesian rationality and no strictly dominant choice is available there. Call a two-person perfect-information game with no indifference among terminal nodes simple, and call it belief-consistent if (a′) and (b′) can be applied to it without logical contradiction — formally (Definition 3) if a JRBS exists for the whole set of relevant decision nodes at once.

Theorem. A simple game is belief-consistent if and only if for every decision node x not on the unique backward induction path, x is irrelevant. That is, either (i) a strictly dominant choice is available at x, or (ii) x is inconsistent with at least one player’s Bayesian rationality.

That is the precise content of “very few.” A simple game is safe only when every off-backward-induction-path node is either trivially decided or already unreachable by rational play — which is why Reny remarks that “all simple games in which player 1 moves first, and player 2 moves second, ending the game, are belief-consistent. This may be one reason for the salience of subgame perfect (i.e., backward induction) outcomes over imperfect Nash outcomes in such games. Of the examples presented here, only TOL(n) for n = 2 and G₁ are belief-consistent.” Two moves deep is fine; three is not — note that this is stronger than the n > 3 in the sentence quoted above, and the two are not in conflict: the earlier statement is about the strong pair (a)/(b), while belief-consistency is about the deliberately weaker (a′)/(b′), whose safe set Reny shows is “no smaller than (and is, in fact, strictly larger than)” the safe set for (a)/(b). The weaker assumptions already fail at TOL(3), because player 2’s node y there is relevant — reachable by Bayesian-rational play, off the backward-induction path, and offering no strictly dominant choice.

Two further details worth carrying, because they are easy to get backwards. First, node-level CBR is not monotone down the tree: in his game G₁ (Figure 4) node y is inconsistent with CBR while its successor z is consistent, “even though z succeeds y” — a player who is surprised is free to revise, and the revision can restore common belief. Second, the relation to subgame perfection runs one way only: “If (s₁*, s₂*) is a subgame perfect equilibrium, then … every node along a subgame perfect equilibrium path is consistent with CBR. The converse, however, fails.”

flowchart TB
    N["node x in a simple<br/>two-person perfect-information game"]
    Q1{"x on the unique<br/>backward induction path?"}
    Q2{"strictly dominant<br/>choice available at x?"}
    Q3{"x consistent with at least<br/>one player's Bayesian rationality?"}
    OK["x is IRRELEVANT<br/>— harmless"]
    BAD["x is RELEVANT off the BI path<br/>→ the game is NOT belief-consistent"]
    N --> Q1
    Q1 -- yes --> OK
    Q1 -- no --> Q2
    Q2 -- yes --> OK
    Q2 -- no --> Q3
    Q3 -- no --> OK
    Q3 -- yes --> BAD
    style BAD fill:#ffecec
    style OK fill:#eefaee

Figure 6b — Reny’s belief-consistency test, node by node. The insight to take away: the theorem is a checklist over off-path nodes, not a global condition, and it fails as soon as one off-path node is both reachable by rational play and genuinely a decision. Since a centipede of length four already has such a node, the class of games where “everyone is rational and everyone knows it” survives contact with the tree is small.

The practical consequence for anyone applying this: sequential equilibrium is where the well-founded part of the refinement programme stops. It has an existence theorem, a clean characterisation, and an implementable definition. Everything above it on the ladder is either equivalent to it generically (perfection), or is a criterion that works on the examples it was designed for and lacks a general justification. In applied work, the honest move is to report the full set of sequential equilibria and then argue informally about plausibility, saying that you are doing so — rather than adopting a refinement whose only credential is that it happens to select the equilibrium you wanted.

Failure Modes and Gotchas

Substituting a uniform belief at unreached information sets. The most common implementation shortcut and the most damaging. It silently converts sequential equilibrium into a different, unnamed concept whose answers happen to agree on symmetric examples and diverge everywhere else. In Selten’s horse, a uniform belief of 1/2 on player 3’s set makes l strictly optimal and destroys the sensible equilibrium — the belief has to be at least 2/3. Make the code refuse to guess.

Forgetting that singleton information sets have forced beliefs even off the path. A distribution over one node is the point mass on it, whether or not the equilibrium reaches it. This is the whole reason Kreps and Wilson’s intermediate criterion can reject (D, a, l) in Selten’s horse with no consistency machinery. Implementations that treat “off the path” as “beliefs unknown” miss this and report the unreasonable equilibrium as fine.

Checking one-shot deviations instead of full continuation plans. The one-shot deviation principle is a theorem about subgame perfection in games with the appropriate continuity, not part of the definition of sequential rationality. Kreps and Wilson’s condition quantifies over all π' differing from π only in the deviator’s component — that is every plan the deviator has, across all his information sets. On games without perfect recall the shortcut is known to fail outright.

Believing that consistency selects sensible beliefs. It does when the zero-probability event has one possible cause (Figure 5: belief pinned to 1/3) and does not when it has several (Selten’s horse: every belief in [0,1] is consistent; Kohlberg’s example: μ(x) = 1/10 is consistent and absurd). Consistency is a coherence condition, not a plausibility condition. Kreps and Wilson wrote an entire section to make this point and still declined to propose a plausibility criterion.

Calling perfect Bayesian equilibrium “sequential.” They are different, several inequivalent definitions of perfect Bayesian are in circulation, and Kreps and Wilson’s own warning identifies the specific casualty: “By dropping (5.4), subgame perfection is lost.” If you weaken consistency far enough, your refinement of subgame perfection stops being one. Concretely: weak perfect Bayesian equilibrium and subgame perfect equilibrium are incomparable — a weak PBE “does not even need to be subgame perfect” — so a paper that says “PBE” and means the weak version has not, in fact, refined subgame perfection at all. And “Bayes’ rule whenever possible” is not a repair, because it is not a definition; González-Díaz and Meléndez-Jiménez exhibit a three-node information set on which two defensible readings of the phrase disagree, and on which even the stronger reading leaves two of the three nodes’ relative weights entirely free. If you must use a perfect Bayesian concept, name which one and state its off-path restriction explicitly.

Assuming a sequential equilibrium is trembling-hand perfect. It generically is, and specifically may not be. The 2×2 example above is sequential and not perfect, and it is not a contrived edge case — the same weak-dominance phenomenon produces 60 of the 61 pure Bayes–Nash equilibria of the small second-price auction computed in Bayesian Games and Types. Whenever weakly dominated strategies are the worry, sequential equilibrium is the wrong tool.

Treating imperfect recall as a technicality. Kreps and Wilson restrict throughout to perfect recall: “In this exposition we consider only games in which each player has perfect recall.” Under imperfect recall the definition itself needs repair, not just the algorithms — see the discussion and the Halpern–Pass citation in Imperfect Information and Information Sets. Abstraction in a game solver is the everyday way of losing perfect recall by accident.

Numerical underflow when verifying a tremble sequence. Realization probabilities at an off-path information set are products of vanishing action probabilities, and in double they underflow to exactly zero long before the belief ratio stops being meaningful. The belief is a ratio of two underflowing quantities, so it is perfectly well conditioned even when both numerator and denominator are denormal. Computing it naively gives 0/0; see the production note below for how the reference implementation avoids this.

Alternatives and When to Choose Them

Concept	What it adds over the one below	Choose it when	Cost
Nash Equilibrium	—	Static, simultaneous, one-shot	Blind off the path entirely
Subgame Perfect Equilibrium	Rules out non-credible threats	The game has genuine proper subgames	Vacuous when it does not — Selten’s horse has zero
Weak perfect Bayesian equilibrium	Beliefs, with Bayes’ rule on the equilibrium path only	Teaching, and as the base case for the stronger notions	Does not imply subgame perfection
Perfect Bayesian equilibrium (Fudenberg–Tirole, Battigalli, “simple”)	Restrictions on off-path beliefs short of full consistency	Applied signalling and industrial-organisation work where the weaker condition suffices; auctions and other continuum-action games where consistency is not even definable	Several inequivalent definitions — always say which; Fudenberg–Tirole’s is only defined for multistage games with observed actions
Sequential equilibrium	Full consistency: beliefs are limits of completely mixed Bayes beliefs	Any game with non-singleton information sets; the default for extensive-form analysis	Consistency is hard to verify; still admits implausible equilibria
Trembling-hand perfect	Robustness of the strategy to the trembles	Weak dominance is the specific worry	Harder to verify; generically the same set
Proper equilibrium (Myerson)	Costlier mistakes are made less often	You want a principled tie-break among trembles	Even harder; a further restriction on the tremble sequence
Strategic stability / forward induction	Set-valued; invariant to some game transformations; captures the Kohlberg argument	You need the “he would not have deviated that way” reasoning	No consensus definition; set-valued answers are awkward to report
Counterfactual Regret Minimization	Not a refinement — a computation that converges to a Nash equilibrium of two-player zero-sum imperfect-information games	The game is too large to solve exactly	Says nothing about beliefs off the equilibrium path; two-player zero-sum only

The selection rule in one paragraph. Count the proper subgames first. If there are enough of them and every node is a singleton, use Backward Induction and stop. If there are proper subgames containing simultaneous moves, use the subgame-perfection fold. If there are no proper subgames worth the name — which is the normal case in signalling games, reputation models, and poker-like games, where Kuhn poker has exactly zero — then subgame perfection is not refining anything and you need beliefs. That is the point at which sequential equilibrium is the right and essentially only well-founded tool. Escalate to trembling-hand perfection only if weak dominance is the specific problem you are trying to rule out, and be aware you are unlikely to change the answer.

Production Notes

How a real solver computes off-path beliefs. The most instructive piece of production code in this area is Gambit’s logit-tracing solver, which computes the agent quantal-response-equilibrium correspondence for extensive games and follows it to a limit point. The relevant file’s header comment states the design problem and the solution in six sentences (src/solvers/logit/logbehav.h, read from the master tree on 2026-08-28):

This is a modified behavior profile class used in the tracing procedure. It assumes that all probabilities are positive; therefore, the log of the probability is well-defined. … This means that beliefs can be computed accurately even for information sets for which the information set realization probability is going to zero. In computing beliefs, for each information set, we find the member reached with maximum probability. … Beliefs are then computed by subtracting the other nodes’ log-realization probability from this “leading” node’s log-realization probability, and then exponentiating the result. This procedure is significant for this application because it is necessary to accurately compute beliefs for information sets whose realization probabilities are going to zero, so as to be able to get a good approximation to the limiting sequential equilibrium.

That is Kreps–Wilson consistency implemented as a numerical method. The class stores m_logProbs alongside m_probs, accumulates log-realization probabilities down the tree, and then applies the log-sum-exp trick per information set:

      T maxLogProb = m_logRealizProbs[infoset->GetMember(1)];
      for (auto member : infoset->GetMembers()) {
        if (m_logRealizProbs[member] > maxLogProb) { maxLogProb = m_logRealizProbs[member]; }
      }
      T total = 0.0;
      for (auto member : infoset->GetMembers()) {
        total += exp(m_logRealizProbs[member] - maxLogProb);
      }
      T mostLikelyBelief = 1.0 / total;
      for (auto member : infoset->GetMembers()) {
        m_beliefs[member] = mostLikelyBelief * exp(m_logRealizProbs[member] - maxLogProb);
      }

Three things to take from this. First, the belief at a vanishing information set is a well-conditioned quantity even when its numerator and denominator both underflow — subtracting the maximum log-probability before exponentiating keeps every term in (0, 1]. Second, the code contains an explicit fallback for total underflow: if the information-set probability evaluates to exactly zero, it sets m_beliefs[member] = 1.0 / infoset->GetMembers().size(), a uniform belief. That is precisely the shortcut warned against above, and its presence here is honest engineering, not sloppiness — it is a last-resort guard on a path the log-space arithmetic is designed never to reach. Third, the phrase “the limiting sequential equilibrium” is the point: the tracing procedure follows a path of completely mixed profiles and its limit is, by construction, a consistent assessment. The algorithm and the definition are the same object.

What the tool actually claims to compute, and on what authority. Gambit ships ten equilibrium-computation programs, and it is worth being clear that nine of them compute Nash equilibria, not sequential ones: the manual’s own framing is “Gambit provides command-line interfaces for each method for computing Nash equilibria” (CLI documentation, read 2026-08-29 against version 17.0.0-alpha.2). Two facts about the current release matter to anyone reproducing this. First, as of 17.0.0 “these programs are now implemented in Python, as part of the pygambit package, rather than being separate C++ executables” — pip install pygambit installs the CLI alongside the library, and there is no separate build. Second, gambit-liap gained a -A switch in 16.5.0 to “report agent Nash equilibria, that is, equilibria which consider only deviations at a single information set at a time”, and since 16.7.0 the old -S switch is gone because the strategic form became the default for extensive games. Agent Nash equilibrium is the concept underlying agent-normal-form perfection discussed above; reading a gambit-liap result as a statement about the tree without -A is a category error the flag exists to prevent.

The single program with a sequential-equilibrium warrant is gambit-logit, and the warrant is a theorem: Theodore Turocy’s Computing sequential equilibria using agent quantal response equilibria (Economic Theory 42(1), 2010, pp. 255–269) states that “the limit of any convergent sequence of agent quantal response equilibria is a sequential equilibrium of an extensive game. Using a logarithmic transformation of action probabilities, it is numerically feasible and practical to compute such sequences, and thereby compute good approximations to sequential equilibrium assessments.” That is the justification for the logbehav.h machinery quoted above — the log-space arithmetic in the code is the “logarithmic transformation of action probabilities” in the theorem. Gambit’s own bibliography lists the paper under the tag [Tur10] (Gambit bibliography).

Uncertain

Verify: the convergence and selection properties Turocy establishes, and the exact hypotheses of the limit theorem. Reason: only the abstract could be retrieved. The publisher copy is behind Springer (10.1007/s00199-009-0443-3), and the author’s own institutional repository record at ueaeprints.uea.ac.uk/id/eprint/19294 says in terms: “Full text not available from this repository.” The quotation above is the author-deposited abstract on that record, which is authoritative for the theorem statement but says nothing about its conditions. To resolve: obtain the Economic Theory article and check whether the limit theorem needs perfect recall, genericity, or a restriction on the branch followed.

None of the reference tools are installable on this machine. Verified again 2026-08-29: no pip, no pip3, no numpy, no gambit-lcp (or any other gambit-*) executable, no pygambit module, no nashpy, no glpk. python3 is present with the standard library only. Gambit and OpenSpiel are readable as source and are worth reading — Gambit’s catalog/ directory in particular turns out to be a small archive of classic examples in machine-readable form, and it is a better citation for a scanned figure’s payoffs than the scan. Everything computed in this note uses fractions.Fraction from the standard library, which for games at whiteboard scale is a better oracle than a floating-point solver because indifference is exactly what sequential rationality turns on.

The concept’s first real application was reputation, and it was the authors’ own. Kreps and Wilson’s companion paper on the chain-store paradox adds “a ‘small’ amount of imperfect (or incomplete) information about players’ payoffs” to a game whose subgame-perfect prediction is that a monopolist never fights entry, and recovers the reputation effect that firms actually display. The self-citation in Sequential Equilibria is telling about how the concept is meant to be used: they point to the companion paper as a case “where there is unique along-the-equilibrium-path behavior among all sequential equilibria whose beliefs meet an intuitively plausible monotonicity condition.” That is the working pattern — enumerate the sequential equilibria, then impose an informal, game-specific belief restriction, and say that you have done so. The type uncertainty that makes it work is set up in Bayesian Games and Types.

A note on provenance — both archive PDFs re-verified 2026-08-29. dklevine.com/archive/ uses opaque numeric filenames that carry no bibliographic information, and the archive has been observed elsewhere in this vault to serve a paper other than the one a citation claimed. Both files this note depends on were therefore re-downloaded in full and identified from their own first pages, not from their filenames or from the citing text:

URL	Received / Content-Length	Identified as
dklevine.com/archive/refs4237.pdf	659,095 / 659,095 bytes, 33 pages	JSTOR scan of Kreps & Wilson, “Sequential Equilibria,” Econometrica 50(4), Jul. 1982, pp. 863–894; PDF Title metadata reads Sequential Equilibria; JSTOR stable URL on page 1 is jstor.org/stable/1912767
dklevine.com/archive/refs4386.pdf	831,893 / 831,893 bytes, 18 pages	Reny, “Common Belief and the Theory of Games with Perfect Information,” Journal of Economic Theory 59, 1993, pp. 257–274; running head “COMMON BELIEF IN GAMES”; received 9 March 1990, revised 20 February 1992

Both citations are accurate. Two operational notes for anyone repeating this. The server has served these files at wildly varying rates — a prior attempt in this vault stalled at roughly 20 KB/min and truncated both downloads — and a truncated PDF produces Syntax Error: Couldn't find trailer dictionary from pdftotext, which reads like corruption but is not: always compare received bytes against Content-Length before concluding a file is bad. And refs4386.pdf is RC4-encrypted (copy permitted) and has no text layer at all — pdffonts lists no embedded fonts, pdftotext yields 18 characters — so it must be read with pdftoppm -png -r 135. Failure to extract is not failure to retrieve.

The Kreps and Wilson text layer is good: 103,858 characters of extractable prose, including all definitions, all propositions, and all discussion. The figures are a different matter — they are scanned line art whose payoff columns OCR into ambiguous blocks. Every figure payoff quoted in this note was reconstructed and then cross-checked against multiple independent quantitative claims in the legible running text, and Selten’s horse was additionally confirmed against Gambit’s transcription. Where a reconstruction could not be confirmed that way, it is not quoted.

See Also

• Subgame Perfect Equilibrium — the rung below, and the note that names this one as the concept that takes over when the subgame supply runs out. It develops the credible-threat argument, the one-shot deviation principle, and the Selten provenance; this note develops the beliefs
• Imperfect Information and Information Sets — why the subgame supply runs out in the first place; the measured result that Kuhn poker has 61 nodes and exactly zero proper subgames, and the perfect-recall hypothesis this concept depends on
• Bayesian Games and Types — the sibling note written alongside this one: where the private information comes from, and why every pooling equilibrium in a signalling game is a claim about off-path beliefs that only this note can adjudicate
• Extensive-Form Games — trees, chance moves, behaviour strategies, and the realization-probability machinery the checker above implements
• Nash Equilibrium — the base of the ladder; Kreps and Wilson’s corollary that in a normal-form game every Nash equilibrium is sequential is what makes the weak-dominance example work
• Backward Induction — the perfect-information special case, where beliefs are trivial and the fold is linear
• Common Knowledge and Rationality Assumptions — the epistemic backdrop; consistency’s upper rungs are explicitly justified by “common knowledge in secondary hypotheses”
• Mixed Strategies — the indifference principle, which is what the 3/4 in Selten’s horse is an instance of
• Counterfactual Regret Minimization — the algorithm that operates on information sets rather than subgames, for the same structural reason this concept does
• Stackelberg Games and Commitment — commitment under private information, where the credibility of a threat becomes a belief question rather than a subgame question
• Games and Strategic Systems in C MOC — the parent map; this note closes the gap between stage P3 (game trees) and stage P6 (learning in imperfect-information games)