#
Exploring the Gap between Perfect Bayesian Equilibrium and Sequential Equilibrium^{ †}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Perfect Bayesian Equilibrium and Sequential Equilibrium

- A finite set of actions A.
- A finite set of histories $H\subseteq {A}^{*}$ which is closed under prefixes (that is, if $h\in H$ and ${h}^{\prime}\in {A}^{*}$ is a prefix of h, then ${h}^{\prime}\in H$). The null history $\u2329\u232a,$ denoted by ∅, is an element of H and is a prefix of every history. A history $h\in H$ such that, for every $a\in A$, $ha\notin H$, is called a terminal history. The set of terminal histories is denoted by Z. $D=H\backslash Z$ denotes the set of non-terminal or decision histories. For every history $h\in H$, we denote by $A\left(h\right)$ the set of actions available at h, that is, $A\left(h\right)=\{a\in A:ha\in H\}$. Thus $A\left(h\right)\ne \u2300$ if and only if $h\in D$. We assume that $A={\bigcup}_{h\in D}A\left(h\right)$ (that is, we restrict attention to actions that are available at some decision history).
- A finite set $N=\{1,\dots ,n\}$ of players. In some cases there is also an additional, fictitious, player called chance.
- A function $\iota :D\to N\cup \left\{chance\right\}$ that assigns a player to each decision history. Thus $\iota \left(h\right)$ is the player who moves at history h. A game is said to be without chance moves if $\iota \left(h\right)\in N$ for every $h\in D.$ For every $i\in N\cup \left\{chance\right\}$, let ${D}_{i}={\iota}^{-1}\left(i\right)$ be the set of histories assigned to player i. Thus $\{{D}_{chance},{D}_{1},\dots ,{D}_{n}\}$ is a partition of $D.$ If history h is assigned to chance, then a probability distribution over $A\left(h\right)$ is given that assigns positive probability to every $a\in A\left(h\right)$.
- For every player $i\in N$, ${\approx}_{i}$ is an equivalence relation on ${D}_{i}$. The interpretation of $h{\approx}_{i}{h}^{\prime}$ is that, when choosing an action at history h, player i does not know whether she is moving at h or at ${h}^{\prime}$. The equivalence class of $h\in {D}_{i}$ is denoted by ${I}_{i}\left(h\right)$ and is called an information set of player i; thus ${I}_{i}\left(h\right)=\{{h}^{\prime}\in {D}_{i}:{h}^{\prime}{\approx}_{i}h\}$. The following restriction applies: if ${h}^{\prime}\in {I}_{i}\left(h\right)$ then $A\left({h}^{\prime}\right)=A\left(h\right)$, that is, the set of actions available to a player is the same at any two histories that belong to the same information set of that player.
- The following property, known as perfect recall, is assumed: for every player $i\in N$, if ${h}_{1},{h}_{2}\in {D}_{i}$, $a\in A\left({h}_{1}\right)$ and ${h}_{1}a$ is a prefix of ${h}_{2}$ then for every ${h}^{\prime}\in {I}_{i}\left({h}_{2}\right)$ there exists an $h\in {I}_{i}\left({h}_{1}\right)$ such that $ha$ is a prefix of ${h}^{\prime}$. Intuitively, perfect recall requires a player to remember what she knew in the past and what actions she took previously.

**Definition**

**1.**

$PL1.$ | $h\precsim ha,\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\forall a\in A\left(h\right)$, |

$PL2.$ | (i) $\phantom{\rule{4pt}{0ex}}\exists a\in A\left(h\right)$ such that $h\sim ha$, |

(ii) $\forall a\in A\left(h\right),$ if $h\sim ha$ then, $\forall {h}^{\prime}\in I\left(h\right)$, ${h}^{\prime}\sim {h}^{\prime}a,$ | |

$PL3.$ | if history h is assigned to chance, then $h\sim ha$, $\forall a\in A\left(h\right).$ |

**Definition**

**2.**

- (i)
- the actions that are assigned positive probability by σ are precisely the plausibility-preserving actions: $\forall h\in D,\forall a\in A\left(h\right)$,$$\sigma \left(a\right)>0\phantom{\rule{4.pt}{0ex}}\mathit{if}\phantom{\rule{4.pt}{0ex}}\mathit{and}\phantom{\rule{4.pt}{0ex}}\mathit{only}\phantom{\rule{4.pt}{0ex}}\mathit{if}\phantom{\rule{4.pt}{0ex}}h\sim ha,$$
- (ii)
- the histories that are assigned positive probability by μ are precisely those that are most plausible within the corresponding information set: $\forall h\in D,$$$\mu \left(h\right)>0\phantom{\rule{4.pt}{0ex}}\mathit{if}\phantom{\rule{4.pt}{0ex}}\mathit{and}\phantom{\rule{4.pt}{0ex}}\mathit{only}\phantom{\rule{4.pt}{0ex}}\mathit{if}\phantom{\rule{4.pt}{0ex}}h\precsim {h}^{\prime},\forall {h}^{\prime}\in I\left(h\right).$$

**Definition**

**3.**

**Definition**

**4.**

**Remark**

**1.**

**Definition**

**5.**

**Definition**

**6.**

**Definition**

**7.**

**Proposition**

**1.**

- (I)
- ($\sigma ,\mu $) is a perfect Bayesian equilibrium which is rationalized by a choicemeasurable plausibility order and is uniformly Bayesian relative to it.
- (II)
- ($\sigma ,\mu $) is a sequential equilibrium.

## 3. Exploring the Gap between PBE and Sequential Equilibrium

**Definition**

**8.**

**Lemma**

**1.**

- (A)
- F satisfies Property $CM$ (Definition 6)
- (B)
- F satisfies the following property: for all $h,{h}^{\prime}\in H$ and $a,b\in A\left(h\right)$, if ${h}^{\prime}\in I\left(h\right)$ then$$F\left(hb\right)-F\left(ha\right)=F\left({h}^{\prime}b\right)-F\left({h}^{\prime}a\right)$$

**Definition**

**9.**

**Proposition**

**2.**

- $Lm\prec Rr$ (because they belong to the same information set and $\mu \left(Lm\right)>0$ while $\mu \left(Rr\right)=0$). Thus if F is any integer-valued representation of ≾ it must be that$$F\left(Lm\right)<F\left(Rr\right).$$
- $Mr\prec L\ell \sim L$ (because $Mr$ and $L\ell $ belong to the same information set and $\mu \left(Mr\right)>0$ while $\mu \left(L\ell \right)=0$; furthermore, ℓ is a plausibility-preserving action since $\sigma \left(\ell \right)>0$). Thus if F is any integer-valued representation of ≾ it must be that$$F\left(Mr\right)<F\left(L\right).$$
- $R\sim R\ell \prec Mm$ (because ℓ is a plausibility-preserving action, $R\ell $ and $Mm$ belong to the same information set and $\mu \left(R\ell \right)>0$ while $\mu \left(Mm\right)=0$). Thus if F is any integer-valued representation of ≾ it must be that$$F\left(R\right)<F\left(Mm\right).$$

## 4. How to Determine if a Plausibility Order Is Choice Measurable

**Remark**

**2.**

**Problem**

**1.**

**Definition**

**10.**

**Proposition**

**3.**

- (A)
- There is a function $F:S\to \mathbb{N}$ such that, for all $s,t,x,y\in S$, (1) $F\left(s\right)\le F\left(t\right)$ if and only if $s\precsim t$; and (2) if $\left(\right[s],[t\left]\right)\doteq \left(\right[x],[y\left]\right)$, with $s\prec t$ and $x\prec y$, then $F\left(t\right)-F\left(s\right)=F\left(y\right)-F\left(x\right)$,
- (B)
- The system of equations corresponding to ≐ (Definition 10) has a solution consisting of positive integers.
- (C)
- There is no sequence $\u2329\left(({i}_{1},{j}_{1})\doteq ({k}_{1},{\ell}_{1})\right),\dots ,\left(({i}_{m},{j}_{m})\doteq ({k}_{m},{\ell}_{m})\right)\u232a$ in ≐ (expressed in terms of the canonical representation ρ of ≾ ) such that ${B}_{left}\u228f{B}_{right}$ where ${B}_{left}={B}_{({i}_{1},{j}_{1})}\u22d3\dots \u22d3{B}_{({i}_{m},{j}_{m})}$ and ${B}_{right}={B}_{({k}_{1},{\ell}_{1})}\u22d3\dots \u22d3{B}_{({k}_{m},{\ell}_{m})}$.

**Remark**

**3.**

## 5. Related Literature

## 6. Conclusions

## Conflicts of Interest

## Appendix A. Proofs

**Proof**

**of**

**Lemma**

**1.**

**Proof**

**of**

**Proposition**

**2.**

**Proof**

**of**

**Proposition**

**3.**

**Lemma**

**2.**

**Proof**.

**Completion**

**of**

**Proof**

**of**

**Proposition**

**3.**

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^{1}The acronym ‘AGM’ stands for ‘Alchourrón-Gärdenfors-Makinson’ who pioneered the literature on belief revision: see [6]. As shown in [7], AGM-consistency can be derived from the primitive concept of a player’s epistemic state, which encodes the player’s initial beliefs and her disposition to revise those beliefs upon receiving (possibly unexpected) information. The existence of a plausibility order that rationalizes the epistemic state of each player guarantees that the belief revision policy of each player satisfies the so-called AGM axioms for rational belief revision, which were introduced in [6].^{2}$\forall h,{h}^{\prime}\in H$, either $h\precsim {h}^{\prime}$ or ${h}^{\prime}\precsim h$.^{3}$\forall h,{h}^{\prime},{h}^{\prime \prime}\in H$, if $h\precsim {h}^{\prime}$ and ${h}^{\prime}\precsim {h}^{\prime \prime}$ then $h\precsim {h}^{\prime \prime}$.^{4}As in [5] we use the notation $h\precsim {h}^{\prime}$ rather than the, perhaps more natural, notation $h\succsim {h}^{\prime}$, for two reasons: (1) it is the standard notation in the extensive literature that deals with AGM belief revision (for a recent survey of this literature see the special issue of the Journal of Philosophical Logic, Vol. 40 (2), April 2011); and (2) when representing the order ≾ numerically it is convenient to assign lower values to more plausible histories. An alternative reading of $h\precsim {h}^{\prime}$ is “history h (weakly) precedes ${h}^{\prime}$ in terms of plausibility”.^{5}A behavior strategy profile is a list of probability distributions, one for every information set, over the actions available at that information set. A system of beliefs is a collection of probability distributions, one for every information set, over the histories in that information set.^{6}The precise definition is as follows. Let Z denote the set of terminal histories and, for every player i, let ${U}_{i}:Z\to \mathbb{R}$ be player i’s von Neumann-Morgenstern utility function. Given a decision history h, let $Z\left(h\right)$ be the set of terminal histories that have h as a prefix. Let ${\mathbb{P}}_{h,\sigma}$ be the probability distribution over $Z\left(h\right)$ induced by the strategy profile σ, starting from history h (that is, if z is a terminal history and $z=h{a}_{1}\dots {a}_{m}$ then ${\mathbb{P}}_{h,\sigma}\left(z\right)={\prod}_{j=1}^{m}\sigma \left({a}_{j}\right)$). Let I be an information set of player i and let ${u}_{i}\left(I\right|\sigma ,\mu )={\displaystyle \sum _{h\in I}}\mu \left(h\right){\displaystyle \sum _{z\in Z\left(h\right)}}{\mathbb{P}}_{h,\sigma}\left(z\right){U}_{i}\left(z\right)$ be player i’s expected utility at I if σ is played, given her beliefs at I (as specified by μ). We say that player i’s strategy ${\sigma}_{i}$ is sequentially rational at I if ${u}_{i}\left(I\right|({\sigma}_{i},{\sigma}_{-i}),\mu )\ge {u}_{i}\left(I\right|({\tau}_{i},{\sigma}_{-i}),\mu )$ for every strategy ${\tau}_{i}$ of player i (where ${\sigma}_{-i}$ denotes the strategy profile of the players other than i). An assessment $(\sigma ,\mu )$ is sequentially rational if, for every player i and for every information set I of player i, ${\sigma}_{i}$ is sequentially rational at $I.\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}$Note that there are two definitions of sequential rationality: the weakly localone—which is the one adopted here—according to which at an information set a player can contemplate changing her choice not only there but possibly also at subsequent information sets of hers, and a strictly local one, according to which at an information set a player contemplates changing her choice only there. If the definition of perfect Bayesian equilibrium (Definition 4 below) is modified by using the strictly local definition of sequential rationality, then an extra condition needs to be added, namely the “pre-consistency” condition identified in [10,11] as being necessary and sufficient for the equivalence of the two notions. For simplicity we have chosen the weakly local definition.^{7}Rounded rectangles represent information sets and the payoffs are listed in the following order: Player 1’s payoff at the top, Player 2’s payoff in the middle and Player 3’s payoff at the bottom.^{8}We use the following convention to represent a total pre-order: if the row to which history h belongs is above the row to which ${h}^{\prime}$ belongs, then $h\prec {h}^{\prime}$ (h is more plausible than ${h}^{\prime}$) and if h and ${h}^{\prime}$ belong to the same row then $h\sim {h}^{\prime}$ (h is as plausible as ${h}^{\prime}$). ∅ denotes the empty history, which corresponds to the root of the tree. In (1) the plausibility-preserving actions are d, e and g; the most plausible histories in the information set $\{a,b,c\}$ are b and c and the two histories in the information set $\{af,bf\}$ are equally plausible.^{9}Given σ, for Player 1 d yields a payoff of 2 while a and c yield 1 and b yields 2; thus d is sequentially rational. Given σ and μ, at her information set $\{a,b,c\}$ with e Player 2 obtains an expected payoff of 4 while with f her expected payoff is 3; thus e is sequentially rational. Given μ, at his information set $\{af,bf\}$, Player 3’s expected payoff from playing with g is 1.5 while his expected payoff from playing with k is 1; thus g is sequentially rational.^{10}Note that if $h,{h}^{\prime}\in E$ and ${h}^{\prime}=h{a}_{1}\dots {a}_{m}$, then $\sigma \left({a}_{j}\right)>0$, for all $j=1,\dots ,m$. In fact, since ${h}^{\prime}\sim h$, every action ${a}_{j}$ is plausibility preserving and therefore, by Property $P1$ of Definition 2, $\sigma \left({a}_{j}\right)>0$.^{11}For an interpretation of the probabilities ${\nu}_{E}\left(h\right)$ see [8].^{12}That is, for every $h\in D\backslash \{\varnothing \}$, ${\mu}^{m}\left(h\right)=\frac{{\displaystyle \prod _{a\in {A}_{h}}}{\sigma}^{m}\left(a\right)}{{\displaystyle \sum _{{h}^{\prime}\in I\left(h\right)}}{\displaystyle \prod _{a\in {A}_{{h}^{\prime}}}}{\sigma}^{m}\left(a\right)}$ (where ${A}_{h}$ is the set of actions that occur in history h). Since ${\sigma}^{m}$ is completely mixed, ${\sigma}^{m}\left(a\right)>0$ for every $a\in A$ and thus ${\mu}^{m}\left(h\right)>0$ for all $h\in D\backslash \{\varnothing \}.$^{13}Since H is finite, there is an $m\in \mathbb{N}$ such that $\{{H}_{0},\dots ,{H}_{m}\}$ is a partition of H and, for every $j,k\in \mathbb{N}$, with $j<k\le m$, and for every $h,{h}^{\prime}\in H$, if $h\in {H}_{j}$ and ${h}^{\prime}\in {H}_{k}$ then $h\prec {h}^{\prime}$.^{14}For example, [12] adopts this interpretation.^{15}For such an interpretation see [6].^{16}Note, however, that $IN{D}_{1}$ is compatible with the following: $a\prec b$ (with $b\in I\left(a\right)$) and $bc\prec ad$ (with $bc\in \phantom{\rule{3.33333pt}{0ex}}I\left(ad\right),\phantom{\rule{4pt}{0ex}}c,d\in \phantom{\rule{3.33333pt}{0ex}}A\left(a\right),\phantom{\rule{4pt}{0ex}}c\ne d$).^{17}We have that (1) $b\prec a$, $bd\prec ad,be\prec ae$ and $bf\prec af$, (2) $ae\prec af$, $aeg\prec afg$ and $aek\prec afk$, (3) $bf\prec be$, $bf\ell \prec be\ell $ and $bfm\prec bem$.^{18}That $IN{D}_{1}$ is satisfied was shown in Footnote 17. $IN{D}_{2}$ is violated because $b\in I\left(a\right)$ and $bf\prec be$ but $ae\prec af$.^{19}In fact, (1) $M\prec L$ and $Mx\prec Lx$ for every $x\in \{\ell ,m,r\}$; (2) $M\prec R$ and $Mx\prec Rx$ for every $x\in \{\ell ,m,r\}$; (3) $R\prec L$ and $Rx\prec Lx$ for every $x\in \{\ell ,m,r\}$; (4) $Mr\prec L\ell $ and $Mrx\prec L\ell x$ for every $x\in \{a,b\}$; (5) $Lm\prec Rr$ and $Lmx\prec Rrx$ for every $x\in \{c,d\}$; and (6) $R\ell \prec Mm$ and $R\ell x\prec Mmx$ for every $x\in \{e,f\}$.^{20}This is easily verified: the important observation is that $Mm\prec Mr$ and $Lm\prec Lr$ and $Rm\prec Rr$. The other comparisons involve a plausibility-preserving action versus a non-plausibility-preserving action and thus $IN{D}_{2}$ is trivially satisfied.^{21}As uniform full support common prior one can take, for example, the uniform distribution over the set of decision histories. Note that, for every equivalence class E of the order, $E\cap \phantom{\rule{4pt}{0ex}}{D}_{\mu}^{+}$ is either empty or a singleton.^{22}To prove that $(\sigma ,\mu )$ is not a sequential equilibrium it is not sufficient to show that plausibility order (5) is not choice measurable, because there could be another plausibility order which is choice measurable and rationalizes $(\sigma ,\mu )$.^{23}As in Definition 5, let ${S}_{0}=\{s\in S:s\precsim t,\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{4pt}{0ex}}\forall t\in S\}$, and, for every integer $k\ge 1$, ${S}_{k}=\{h\in S\phantom{\rule{3.33333pt}{0ex}}\backslash \phantom{\rule{3.33333pt}{0ex}}{S}_{0}\cup \dots \cup \phantom{\rule{3.33333pt}{0ex}}{S}_{k-1}:\phantom{\rule{3.33333pt}{0ex}}s\precsim t,\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{4pt}{0ex}}\forall t\in S\phantom{\rule{3.33333pt}{0ex}}\backslash \phantom{\rule{3.33333pt}{0ex}}{S}_{0}\cup \dots \cup \phantom{\rule{3.33333pt}{0ex}}{S}_{k-1}\}$. The canonical ordinal integer-valued representation of ≾, $\rho :S\to \mathbb{N}$, is given by $\rho \left(s\right)=k$ if and only if $s\in {S}_{k}.$^{24}Thus $a\prec x$ for every $x\in S\backslash \left\{a\right\}$, $\left[b\right]=\{b,c\}$, $b\prec d$, etc.^{25}For example, ≐ is the smallest reflexive, symmetric and transitive relation that contains the pairs given in (14).^{26}The system of linear equations of Definition 10 is somewhat related to the system of multiplicative equations considered in [13] (Theorem 5.1). A direct comparison is beyond the scope of this paper and is not straightforward, because the structures considered in Definition 10 are more general than those considered in [13].^{27}By symmetry of ≐, we can express the third and fourth constraints as $(4,6)\doteq (0,2)$ and $(3,4)\doteq (1,3)$ instead of $(0,2)\doteq (4,6)$ and $(1,3)\doteq (3,4)$, respectively.^{28}The main element of the notion of PBE put forward in [16] is the “no signaling what you don’t know” condition on beliefs. For example, if Player 2 observes Player 1’s action and Player 1 has observed nothing about a particular move of Nature, then Player 2 should not update her beliefs about Nature’s choice based on Player 1’s action.^{29}Intuitively, on consecutive information sets, a player does not change her beliefs about the actions of other players, if she has not received information about those actions.^{30}By “Bayesian updating as long as possible” we mean the following: (1) when information causes no surprises, because the play of the game is consistent with the most plausible play(s) (that is, when information sets are reached that have positive prior probability), then beliefs should be updated using Bayes’ rule; and (2) when information is surprising (that is, when an information set is reached that had zero prior probability) then new beliefs can be formed in an arbitrary way, but from then on Bayes’ rule should be used to update those new beliefs, whenever further information is received that is consistent with those beliefs.^{31}It is straightforward to check that if ${F}^{\prime}:H\to \mathbb{N}$ is an integer-valued representation of ≾ then so is $F:H\to \mathbb{N}$ defined by $F\left(h\right)={F}^{\prime}\left(h\right)-{F}^{\prime}(\varnothing )$; furthermore if ${F}^{\prime}$ satisfies property $CM$ ($CM\prime $) then so does F.^{32}For example, if $S=\{a,b,c,d,e,f\}$ and ≾ is given by $a\sim b\prec c\prec d\sim e\prec f$ then $\rho \left(a\right)=\rho \left(b\right)=0,\rho \left(c\right)=1,\rho \left(d\right)=\rho \left(e\right)=2$ and $\rho \left(f\right)=3$; if F is given by $F\left(a\right)=F\left(b\right)=0,F\left(c\right)=3,F\left(d\right)=F\left(e\right)=5$ and $F\left(f\right)=9$ then ${\widehat{x}}_{0}=0,{\widehat{x}}_{1}=3,{\widehat{x}}_{2}=2$ and ${\widehat{x}}_{3}=4$.^{33}For example, the system of Equation (15) can be written as $Ax=0$, where $x=({x}_{1},\dots ,{x}_{5})$ and$$A=\left(\begin{array}{ccccc}\hfill 1& \hfill 1& \hfill 0& \hfill -1& \hfill 0\\ \hfill -1& \hfill -1& \hfill 0& \hfill 1& \hfill 0\\ \hfill 0& \hfill 0& \hfill 1& \hfill 0& \hfill -1\\ \hfill 0& \hfill 0& \hfill -1& \hfill 0& \hfill 1\\ \hfill 1& \hfill 1& \hfill 1& \hfill -1& \hfill -1\\ \hfill -1& \hfill -1& \hfill -1& \hfill 1& \hfill 1\end{array}\right)$$^{35}Proof. Recall that for each row ${a}_{i}$ of A there is a row ${a}_{k}$ such that ${a}_{i}=-{a}_{k}$. If ${y}_{i}\ne 0$ and ${y}_{k}\ne 0$ for some i and k such that ${a}_{i}=-{a}_{k}$ then$${y}_{i}{a}_{i}+{y}_{k}{a}_{k}=\left\{\begin{array}{cc}0& \mathrm{i}\mathrm{f}\text{}{y}_{i}={y}_{k}\\ \hfill ({y}_{k}-{y}_{i}){a}_{k}& \mathrm{i}\mathrm{f}\text{}0{y}_{i}{y}_{k}\\ \hfill ({y}_{i}-{y}_{k}){a}_{i}& \mathrm{i}\mathrm{f}\text{}0{y}_{k}{y}_{i}\\ \hfill \left(\left|{y}_{i}\right|+{y}_{k}\right){a}_{k}& \mathrm{i}\mathrm{f}\text{}{y}_{i}0{y}_{k}\\ \hfill \left({y}_{i}+\left|{y}_{k}\right|\right){a}_{i}& \mathrm{i}\mathrm{f}\text{}{y}_{k}0{y}_{i}\\ \hfill \left(\left|{y}_{k}\right|-\left|{y}_{i}\right|\right){a}_{i}& \mathrm{i}\mathrm{f}\text{}{y}_{i}{y}_{k}0\\ \hfill \left(\left|{y}_{i}\right|-\left|{y}_{k}\right|\right){a}_{k}& \mathrm{i}\mathrm{f}\text{}{y}_{k}{y}_{i}0\end{array}\right.$$^{36}Proof. Suppose that ${y}_{k}<0$ for some $k\in K$. Recall that there exists an i such that ${a}_{k}=-{a}_{i}$. By the argument of the previous footnote, ${y}_{i}=0$. Then replace ${y}_{k}$ by 0 and replace ${y}_{i}=0$ by ${\tilde{y}}_{i}=-{y}_{k}$.^{37}For example, if $K=\{3,6,7\}$ and ${y}_{3}=2$, ${y}_{6}=1$, ${y}_{7}=3$, then B is the $3\times n$ matrix where ${b}_{1}={a}_{3},\phantom{\rule{3.33333pt}{0ex}}{b}_{2}={a}_{6}$ and ${b}_{3}={a}_{7}$ and ${\alpha}_{1}=2$, ${\alpha}_{2}=1$ and ${\alpha}_{3}=3$.

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Bonanno, G. Exploring the Gap between Perfect Bayesian Equilibrium and Sequential Equilibrium. *Games* **2016**, *7*, 35.
https://doi.org/10.3390/g7040035

**AMA Style**

Bonanno G. Exploring the Gap between Perfect Bayesian Equilibrium and Sequential Equilibrium. *Games*. 2016; 7(4):35.
https://doi.org/10.3390/g7040035

**Chicago/Turabian Style**

Bonanno, Giacomo. 2016. "Exploring the Gap between Perfect Bayesian Equilibrium and Sequential Equilibrium" *Games* 7, no. 4: 35.
https://doi.org/10.3390/g7040035