# Rational Play in Extensive-Form Games

## Abstract

**:**

## 1. Introduction

## 2. What Is a Rational Solution?

**1. The Nash-equilibrium approach.**Within this approach the notion of rationality is captured through the concept of Nash equilibrium or one of its refinements. Consider, for example, the game of Figure 1 and the strategy profile $(a,a,d)$.

**2. The self-confirming equilibrium approach.**Returning to the strategy profile $(a,a,d)$ in the game of Figure 1, an alternative approach is to interpret Player 3’s strategy d not as a claim about what Player 3 would actually do in a counterfactual world where her information is reached, but as a belief, shared by Players 1 and 2, about Player 3’s hypothetical behavior. Such shared belief would support the rationality of playing a for both Players 1 and 2.

## 3. Behavioral Models of Games

#### 3.1. Qualitative Beliefs

#### 3.2. Models of Games

**Definition 1.**

- Ω is a set of states.
- $\zeta :\mathrm{\Omega}\to Z$ is an assignment of a terminal history to each state.
- For every $h\in D$, ${\mathcal{B}}_{h}\subseteq \mathrm{\Omega}\times \mathrm{\Omega}$ is a belief relation that satisfies the following properties:
- 1.
- ${\mathcal{B}}_{h}\left(\omega \right)\ne \u2300$ if and only if $h\prec \zeta \left(\omega \right)$ [beliefs are specified only at reached decision histories and are consistent: consistency means that there is no event $\mathbf{E}$ such that both $\mathbf{E}$ and its complement $\neg \mathbf{E}$ are believed; it is well known that, at state ω, beliefs are consistent if and only if $\mathcal{B}\left(\omega \right)\ne \u2300$].
- 2.
- If ${\omega}^{\prime}\in {\mathcal{B}}_{h}\left(\omega \right)$ then ${h}^{\prime}\prec \zeta \left({\omega}^{\prime}\right)$ for some ${h}^{\prime}$ such that ${h}^{\prime}{\approx}_{\iota \left(h\right)}h$ [the active player at history h correctly believes that her information set that contains h has been reached; recall (see Appendix A) that ${h}^{\prime}{\approx}_{\iota \left(h\right)}h$ (also written as ${h}^{\prime}\in \left[h\right]$) if and only if h and ${h}^{\prime}$ belong to the same information set of player $\iota \left(h\right)$ (thus $\iota \left(h\right)=\iota \left({h}^{\prime}\right))$].
- 3.
- If ${\omega}^{\prime}\in {\mathcal{B}}_{h}\left(\omega \right)$ then (1) ${\mathcal{B}}_{h}\left({\omega}^{\prime}\right)={\mathcal{B}}_{h}\left(\omega \right)$ and (2) if ${h}^{\prime}\prec \zeta \left({\omega}^{\prime}\right)$ with ${h}^{\prime}{\approx}_{\iota \left(h\right)}h$ then ${\mathcal{B}}_{{h}^{\prime}}\left({\omega}^{\prime}\right)={\mathcal{B}}_{h}\left(\omega \right)$ [by (1), beliefs satisfy positive and negative introspection and, by (2), beliefs are the same at any two histories in the same information set; thus one can unambiguously refer to a player’s beliefs at an information set, which is what we do in Figures 5–9].
- 4.
- If ${\omega}^{\prime}\in {\mathcal{B}}_{h}\left(\omega \right)$ and ${h}^{\prime}\prec \zeta \left({\omega}^{\prime}\right)$ with ${h}^{\prime}{\approx}_{\iota \left(h\right)}h$, then, for every action $a\in A\left(h\right)$ (note that $A\left({h}^{\prime}\right)=A\left(h\right)$), there is an ${\omega}^{\u2033}\in {\mathcal{B}}_{h}\left(\omega \right)$ such that ${h}^{\prime}a\precsim \zeta \left({\omega}^{\u2033}\right)$.

- 1.
- As a matter of fact, Player 1 plays a, Player 2 plays b and Player 3 plays d.
- 2.
- Player 1 (who chooses at the null history ⌀) believes that if she plays a then Player 2 will also play a (this belief is erroneous since at state $\gamma $ Player 2 actually plays b, after Player 1 plays a) and thus her utility will be 2, and she believes that if she plays b then Player 2 will play a and Player 3 will play d and thus her utility will be 1.
- 3.
- Player 2 (who chooses at information set $\{a,b\}$) correctly believes that Player 1 played a and, furthermore, correctly believes that if he plays b then Player 3 will play d and thus his utility will be 1, and believes that if he plays a his utility will be 2.
- 4.
- Player 3 (who chooses at information set $\{ab,ba,bb\}$) erroneously believes that both Player 1 and Player 2 played b; thus, she believes that if she plays c her utility will be 0 and if she plays d her utility will be 1.

#### 3.3. Rationality

**active**players are rational, then in the model of Figure 5 we have that $\mathbf{R}=\left\{\beta \right\}$ (note that at state $\beta $ Player 3 is not active).

**Definition 2.**

**(A)**- We say that, at ω and h, the active player $\iota \left(h\right)$ believes that b is better than a if, for all ${\omega}_{1},{\omega}_{2}\in {\mathcal{B}}_{h}\left(\omega \right)$ and for all ${h}^{\prime}$ such that ${h}^{\prime}{\approx}_{\iota \left(h\right)}h$ (that is, history ${h}^{\prime}$ belongs to the same information set as h), if a is the action taken at history ${h}^{\prime}$ at state ${\omega}_{1}$, that is, ${h}^{\prime}a\precsim \zeta \left({\omega}_{1}\right)$, and b is the action taken at ${h}^{\prime}$ at state ${\omega}_{2}$, that is, ${h}^{\prime}b\precsim \zeta \left({\omega}_{2}\right)$, then ${u}_{\iota \left(h\right)}\left(\zeta \left({\omega}_{1}\right)\right)<{u}_{\iota \left(h\right)}\left(\zeta \left({\omega}_{2}\right)\right)$. Thus, the active player at history h believes that action b is better than action a if, restricting attention to the states that she considers possible, the largest utility that she obtains if she plays a is less than the lowest utility that she obtains if she plays b.
**(B)**- We say that player $\iota \left(h\right)$ is rational at history h at state ω if and only if the following is true: if $ha\precsim \zeta \left(\omega \right)$ (that is, $a\in A\left(h\right)$ is the action played at h at state ω) then, for every $b\in A\left(h\right)$, it is not the case that, at state ω and history h, player $\iota \left(h\right)$ believes that b is better than a.

#### 3.4. Correct Beliefs

#### 3.5. Self-Confirming Play

**Definition 3.**

- 1.
- if ${h}^{\prime}\prec \zeta \left({\omega}^{\prime}\right)$ and ${h}^{\u2033}\prec \zeta \left({\omega}^{\u2033}\right)$ then ${h}^{\prime}={h}^{\u2033}$,
- 2.
- $\forall a\in A\left(h\right)$, if ${h}^{\prime}a\prec \zeta \left({\omega}^{\prime}\right)$ and ${h}^{\u2033}a\prec \zeta \left({\omega}^{\u2033}\right)$ then $\zeta \left({\omega}^{\prime}\right)=\zeta \left({\omega}^{\u2033}\right)$.

**Definition 4.**

**Definition 5.**

**Definition 6.**

- 5.
- Let ω be a state, h a decision history reached at ω ($h\prec \zeta \left(\omega \right)$) and a and b two actions available at h ($a,b\in A\left(h\right)$). Let ${h}_{1}$ and ${h}_{2}$ be two decision histories that belong to the same information set of player $j=\iota \left({h}_{1}\right)$ (${h}_{1}{\approx}_{j}{h}_{2}$) and ${c}_{1},{c}_{2}$ be two actions available at ${h}_{1}$ (${c}_{1},{c}_{2}\in A\left({h}_{1}\right)=A\left({h}_{2}\right)$). Then the following holds (recall that $\left[h\right]$ denotes the information set that contains decision history h, that is, ${h}^{\prime}\in \left[h\right]$ if and only if ${h}^{\prime}{\approx}_{\iota \left(h\right)}h$):

**Definition 7.**

- 1.
- two decision histories ${h}_{1}$ and ${h}_{2}$ that are reached at ω (that is, ${h}_{1}\prec {h}_{2}\prec \zeta \left(\omega \right)$) and belong to i and j, respectively, (that is, $i=\iota \left({h}_{1}\right)$ and $j=\iota \left({h}_{2}\right)$),
- 2.
- states ${\omega}_{1}\in {\mathcal{B}}_{{h}_{1}}\left(\omega \right)$ and ${\omega}_{2}\in {\mathcal{B}}_{{h}_{2}}\left(\omega \right)$,
- 3.
- decision histories ${h}^{\prime},{h}^{\u2033}\in \left[h\right]$,

**Definition 8.**

- 1.
- if ${\omega}_{1}\in {\mathcal{B}}_{{h}_{1}}\left(\omega \right)$ and ${h}_{1}^{\prime}\prec {h}^{\prime}a\precsim \zeta \left({\omega}_{1}\right)$ with ${h}_{1}^{\prime}\in \left[{h}_{1}\right]$, ${h}^{\prime}\in \left[h\right]$ and $a\in A\left(h\right)$, then there exists an ${\omega}_{2}\in {\mathcal{B}}_{{h}_{2}}\left(\omega \right)$ such that, for some ${h}^{\u2033}\in \left[h\right]$ and ${h}_{2}^{\prime}\in \left[{h}_{2}\right]$, ${h}_{2}^{\prime}\prec {h}^{\u2033}a\precsim \zeta \left({\omega}_{2}\right)$, and
- 2.
- if ${\omega}_{2}\in {\mathcal{B}}_{{h}_{2}}\left(\omega \right)$ with ${h}_{2}^{\prime}\prec {h}^{\u2033}b\precsim \zeta \left({\omega}_{2}\right)$ with ${h}_{2}^{\prime}\in \left[{h}_{2}\right]$, ${h}^{\u2033}\in \left[h\right]$ and $b\in A\left(h\right)$ then here exists an ${\omega}_{1}\in {\mathcal{B}}_{{h}_{1}}\left(\omega \right)$ such that, for some ${h}^{\prime}\in \left[h\right]$ and ${h}_{1}^{\prime}\in \left[{h}_{1}\right]$, ${h}_{1}^{\prime}\prec {h}^{\prime}b\precsim \zeta \left({\omega}_{1}\right)$.

**Proposition 1.**

**(A)**- If z is a Nash play of G then there is a causally restricted model of G and a state $\omega $ in that model such that (1) $\zeta \left(\omega \right)=z$ and (2) $\omega \in \mathbf{R}\cap \mathbf{T}\cap \mathbf{C}\cap \mathbf{A}$.
**(B)**- For any causally restricted model of G and for every state $\omega $ in that model, if $\omega \in \mathbf{R}\cap \mathbf{T}\cap \mathbf{C}\cap \mathbf{A}$ then $\zeta \left(\omega \right)$ is a Nash play.

## 4. Further Discussion and Conclusions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. The History-Based Definition of Extensive-Form Game

- 1.
- A set of players denoted by N.
- 2.
- A set of actions, denoted by A.
- 3.
- A set of histories, denoted by $H\subseteq {A}^{*}$, which satisfies the property that, if $h\in H$ and ${h}^{\prime}\in {A}^{*}$ is a prefix of h, then ${h}^{\prime}\in H$. The null history $,$ denoted by ⌀, belongs to 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 or play. Z denotes the set of terminal histories and $D=H\setminus Z$ the set of decision histories.
- 4.
- To every decision history is assigned a player, by means of a function $\iota :D\to N$. Thus, $\iota \left(h\right)\in N$ is the player who moves, or is active, at $h\in D$. For notational simplicity we assume that there is exactly one player who is active active at any decision history; thus, a simultaneous move by, say, Players 1 and 2 is represented in the traditional way by having Player 1 move first followed by Player 2, who is not informed of Player 1’s move. Let ${D}_{i}=\{h\in D:i=\iota \left(h\right)\}$ denote the set of histories at which player i is active. For every $h\in D$, $A\left(h\right)$ denotes the set of actions available at h (to player $\iota \left(h\right)$), that is, $a\in A\left(h\right)$ if and only if $a\in A$ and $ha\in H$.
- 5.
- For every player $i\in N$, we postulate an equivalence relation ${\approx}_{i}$ on ${D}_{i}$: $h{\approx}_{i}{h}^{\prime}$ if and only if, when choosing an action at history $h\in {D}_{i}$, player i does not know whether she is moving at h or at ${h}^{\prime}$. The equivalence class of $h\in D$ is denoted by $\left[h\right]$ and is called an information set of player $\iota \left(h\right)$; thus $\left[h\right]=\{{h}^{\prime}\in {D}_{\iota \left(h\right)}:h{\approx}_{\iota \left(h\right)}{h}^{\prime}\}$. The actions available at an information set are not allowed to differ across histories in that information set, that is, if $h{\approx}_{i}{h}^{\prime}$ then $A\left({h}^{\prime}\right)=A\left(h\right)$. We also assume the property of perfect recall, according to which a player always remembers her own past moves: 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 ${h}^{\prime}$ such that ${h}^{\prime}{\approx}_{i}{h}_{2}$, there exists an $h{\approx}_{i}{h}_{1}$ such that $ha$ is a prefix of ${h}^{\prime}$.When every information set consists of a single history, the game is said to have perfect information, otherwise it is said to have imperfect information.

## Appendix B. Proof of Proposition 1

**Definition A1.**

**Definition A2.**

- $\mathrm{\Omega}=Z$.
- $\zeta :Z\to Z$ is the identity function: $\zeta \left(z\right)=z,\forall z\in Z$.
- For every $h\in D$ and $z\in Z$ define ${\mathcal{B}}_{h}\left(z\right)$ as follows:
- 1.
- If $h\nprec z$, then ${\mathcal{B}}_{h}\left(z\right)=\u2300$.
- 2.
- If $h\prec {z}_{s}^{*}$ then ${\mathcal{B}}_{h}\left({z}_{s}^{*}\right)=\left\{{z}^{\prime}\in Z:{z}^{\prime}={f}_{s}\left(ha\right)\phantom{\rule{4.pt}{0ex}}\mathrm{for}\phantom{\rule{4.pt}{0ex}}\mathrm{some}\phantom{\rule{4.pt}{0ex}}a\in A\left(h\right)\right\}$. [That is, if h is on the play generated by s, then at h the active player believes that, for every available action a, if she takes action a then the outcome will be the terminal history reached from $ha$ by s.]
- 3.
- If $h\nprec {z}_{s}^{*}$, but $\left[h\right]$ is not avoided by ${z}_{s}^{*}$, then, for all $z\in Z$ such that $h\prec z$, ${\mathcal{B}}_{h}\left(z\right)=\left\{{z}^{\prime}\in Z:{z}^{\prime}={f}_{s}\left({h}_{s}^{*}\left(\left[h\right]\right)a\right)\phantom{\rule{4.pt}{0ex}}\mathrm{for}\phantom{\rule{4.pt}{0ex}}\mathrm{some}\phantom{\rule{4.pt}{0ex}}a\in A\left(h\right)\right\}$. [That is, at every decision history in an information set crossed by the play generated by s, the player believes that the play has reached history ${h}_{s}^{*}\left(\left[h\right]\right)$ (the history in $\left[h\right]$ that is on the play to ${z}_{s}^{*}$) and her beliefs are as given in Point 2.]
- 4.
- If $\left[h\right]$ is avoided by ${z}_{s}^{*}$, let $\widehat{h}={g}_{s}\left(\left[h\right]\right)$. Then, for every ${h}^{\prime}\in \left[h\right]$ and every $z\in Z$ such that ${h}^{\prime}\prec z$, ${\mathcal{B}}_{{h}^{\prime}}\left(z\right)=\{{z}^{\prime}\in Z:{z}^{\prime}={f}_{s}\left(\widehat{h}a\right)\phantom{\rule{4.pt}{0ex}}\mathrm{for}\phantom{\rule{4.pt}{0ex}}\mathrm{some}\phantom{\rule{4.pt}{0ex}}a\in A\left(h\right)\}$. [That is, at every decision history in an information set that is not crossed by the play generated by s, the player believes that she is at the history selected by ${g}_{s}$, denoted by $\widehat{h}$, and that, for every available action a, if she takes action a then the outcome will be the terminal history reached from $\widehat{h}a$ by s.]

**Remark A1.**

**Remark A2.**

**Proof.**

**(A)**[Note that, for this part of the proof, the restriction that no player moves more than once along any play of the game is not needed.] Fix a finite extensive-form game G and let s be a pure-strategy Nash equilibrium s of G. Fix a selection function ${g}_{s}$ based on s (Definition A1) and consider the model generated by s and ${g}_{s}$ (Definition A2). By Remark A2, ${z}_{s}^{*}\in \mathbf{C}\cap \mathbf{T}\cap \mathbf{A}$ (recall that ${z}_{s}^{*}$ is the play generated by s, that is, ${z}_{s}^{*}={f}_{s}(\u2300)$). Thus, it only remains to show that ${z}_{s}^{*}\in \mathbf{R}$. If h is a decision history, denote by $s\left(h\right)$ the choice selected by s at h. Fix an arbitrary decision history h that is reached at state ${z}_{s}^{*}$ (that is, $h\prec {z}_{s}^{*}$) and let a be the action at h such that $ha\precsim {z}_{s}^{*}$, that is, $s\left(h\right)=a$; then ${f}_{s}\left(ha\right)={f}_{s}(\u2300)={z}_{s}^{*}$. Suppose that player $\iota \left(h\right)$ is not rational at h. Then there must be a $b\in A\left(h\right)\setminus \left\{a\right\}$ that guarantees a higher utility to player $\iota \left(h\right)$: if ${z}^{\prime}\in {\mathcal{B}}_{h}\left({z}_{s}^{*}\right)$ is such that $hb\precsim {z}^{\prime}$, then ${u}_{\iota \left(h\right)}\left({z}^{\prime}\right)>{u}_{\iota \left(h\right)}\left({z}_{s}^{*}\right)$. By Definition A2, ${z}^{\prime}={f}_{s}\left(hb\right)$ so that ${u}_{\iota \left(h\right)}\left({f}_{s}\left(hb\right)\right)>{u}_{\iota \left(h\right)}\left({f}_{s}\left(ha\right)\right)$; hence, by unilaterally changing her strategy at h from a to b (while leaving the rest of her strategy unchanged), player $\iota \left(h\right)$ can increase her payoff, contradicting the assumption that s is a Nash equilibrium.

**(B)**Fix a finite extensive-form game G where no player moves more than once along any play and consider an arbitrary model of it where there is a state $\alpha $ such that $\alpha \in \mathbf{R}\cap \mathbf{T}\cap \mathbf{C}\cap \mathbf{A}$. We want to show that we can construct a pure-strategy Nash equilibrium s of G such that ${f}_{s}(\u2300)=\zeta \left(\alpha \right)$.

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**Figure 2.**The conflict between the backward-induction-based counterfactual and the forward-induction-based counterfactual encoded in Player 2’s strategy.

**Figure 3.**The play $aA$ is consistent with the notion of self-confirming equilibrium, even though there is no Nash equilibrium that yields $aA$.

**Figure 4.**The relation $\mathcal{B}=\left\{\right(\alpha ,\beta ),(\alpha ,\mathrm{\gamma}),(\beta ,\beta ),(\beta ,\mathrm{\gamma}),(\mathrm{\gamma},\beta ),(\mathrm{\gamma},\mathrm{\gamma}\left)\right\}$.

**Figure 5.**The top part reproduces the game of Figure 1 and the bottom part shows a model of it.

**Figure 7.**The game of Figure 3 and a model of it.

**Figure 8.**A game and four partial models of it (showing only the beliefs of Player 1 at history ⌀), two of which violate Condition 5 of Definition 6 and the remaining two do not.

**Figure 9.**$\mathrm{\gamma}\in \mathbf{R}\cap \mathbf{T}\cap \mathbf{C}\cap \mathbf{A}$ but at $\mathrm{\gamma}$ Player 1 does not believe that Player 2 is rational.

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Bonanno, G.
Rational Play in Extensive-Form Games. *Games* **2022**, *13*, 72.
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Bonanno G.
Rational Play in Extensive-Form Games. *Games*. 2022; 13(6):72.
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Bonanno, Giacomo.
2022. "Rational Play in Extensive-Form Games" *Games* 13, no. 6: 72.
https://doi.org/10.3390/g13060072