# Equilibrium selection through \(\mathbf {p}_{u}\)-dominance

## Abstract

The paper introduces and discusses the concept of \(\mathbf {p}_{u}\)-dominance in the context of finite games in normal form. It then presents the \(\mathbf { p}_{u}\)-dominance criterion for equilibrium selection. The \(\mathbf {p}_{u}\)-dominance criterion is inspired by and closely related to the \(\mathbf {p}\)-dominance criterion originally proposed by Morris et al. (Econometrica 63:145–157, 1995). However, there are games in which the two criteria lead to different refinements. We provide sufficient conditions under which equilibrium selection through \(\mathbf {p}_{u}\)-dominance is weakly finer (respectively, coarser) than equilibrium selection through \(\mathbf {p}\)-dominance.

### Keywords

Equilibrium selection Normal form games \(\mathbf {p}_{u}\)-Dominance \(\mathbf {p}\)-Dominance### JEL Classification

C72 C73## 1 Introduction

Multiplicity of equilibria is a feature that characterizes many strategic interactions. A typical example is a coordination game where multiple pure strategy Nash equilibria occur when players’ actions match. Whenever multiple equilibria exist, equilibrium selection obviously becomes a key issue, both from a normative and a positive point of view (Kim 1996; Haruvy and Stahl 2007).

In this paper, we introduce the concept of \(\mathbf {p}_{u}\)-dominance and use it as a criterion for equilibrium selection in the context of finite one-shot simultaneous games. In a nutshell, the \(\mathbf {p}_{u}\)-dominance criterion selects the equilibrium whose supporting actions turn out to be \(\mathbf {p}_{u}\)-dominant for the less stringent set of beliefs. More precisely, we say that an equilibrium is \(\mathbf {p}_{u}\)-dominant for the vector \(\mathbf {p}_{u}=(p_{1},\ldots ,p_{n})\) if, for every player \(i\in \left\{ 1,\ldots ,n\right\} \), \(i\)’s equilibrium action best responds to any conjecture according to which any player \(j\ne i\) plays his equilibrium action with probability of at least \(p_{i}\) and uniformly randomizes the remaining probability over his alternative actions.^{1}

In terms of its “microfoundations”, \( \mathbf {p}_{u}\)-dominance thus mimics the mental process according to which every agent evaluates the likelihood of an equilibrium by focusing on the probability of the actions that sustain it, while assuming a streamlined uniform distribution for what concerns the other players’ alternative actions. Such an approach can be the result of bounded rationality or can stem from a conscious consideration of the player. In this respect, \(\mathbf { p}_{u}\)-dominance postulates a behavior that is consistent with the one that the so-called “wordly” archetype of players would implement (for experimental evidence about the incidence of this archetype of players see Stahl and Wilson 1995; Haruvy and Stahl 2007; Weizsäcker 2003): play best response to a prior based on a belief that a fraction of the population chooses the equilibrium strategy (so called naive-Nash type) while the remaining part uniformly randomizes over all strategies (level-0 type).

Notice also that, by construction, the \(\mathbf {p}_{u}\)-dominance criterion evaluates an equilibrium in light of a conjecture that assigns positive probability to the event that the opponents deviate and do not play their corresponding equilibrium action. As such, \(\mathbf {p}_{u}\)-dominance considers not only the profitability of playing a certain equilibrium action but also its riskiness. The \(\mathbf {p}_{u}\)-dominance criterion thus tackles the issue of equilibrium selection using the same intuition that underlies the well-known concepts of risk-dominance (Harsanyi and Selten 1988) and \(\mathbf {p}\)-dominance (Morris et al. 1995; Kajii and Morris 1997): if multiple equilibria exist and agents do not know which equilibrium will arise, they will coordinate their expectations on the one that better solves the trade-off between risk and return.

In the second part of the paper, we more thoroughly investigate the relationships that exist between \(\mathbf {p}_{u}\)-dominance, risk-dominance, and \(\mathbf {p}\)-dominance. We show that the \(\mathbf {p}_{u}\)-dominance criterion and the risk-dominance criterion coincide in any \(2 \times 2\) games, but \( \mathbf {p}_{u}\)-dominance is a more general concept, as it can also be applied to games that have more than two players. We then compare \(\mathbf {p} _{u}\)-dominance and \(\mathbf {p}\)-dominance. The two concepts are clearly similar in their approach (\(\mathbf {p}_{u}\)-dominance is actually inspired by \(\mathbf {p}\)-dominance) but they may lead to different conclusions in terms of equilibrium selection. We show in fact that there exist games in which the \(\mathbf {p}_{u}\)-dominance criterion leads to a finer selection (i.e., it sensibly discriminates among equilibria that the \(\mathbf {p}\)-dominance criterion deems equal) as well as games in which the opposite holds true. We then provide sufficient conditions that discriminate between the two cases and at the same time ensure that the \(\mathbf {p}_{u}\)-dominance criterion selects a unique equilibrium.

## 2 The concept of \(\mathbf {p}_{u}\)-dominance

Consider a generic one-shot simultaneous game \(G=\left\langle N,\left( A_{i}\right) _{i\in N},\left( u_{i}\right) _{i\in N}\right\rangle \) where \( N=\left\{ 1,\ldots ,n\right\} \) with \(n\ge 2\) denotes the set of players, \( A_{i}=\left\{ a_{1},\ldots ,a_{k_{i}}\right\} \) with \(k_{i}\ge 2\) is the action space of player \(i\in N\), and \(u_{i}:A\rightarrow \mathbb {R} \) is the payoff function with \(A=\times _{j\in N}A_{j}\). As usual, a profile \(a^{*}\in A\) is a Nash equilibrium of \(G\) if the relation \( u_{i}(a_{i}^{*},a_{-i}^{*})\ge u_{i}\left( a_{i},a_{-i}^{*}\right) \) holds for every \(i\in N\) and every \(a_{i}\in A_{i}\).

Now take any action profile \(\hat{a}\) and define the vector \(\mathbf {p}_{u}( \hat{a})=\left( p_{1}(\hat{a}),\ldots ,p_{n}(\hat{a})\right) \). The vector \( \mathbf {p}_{u}(\hat{a})\) assigns to every player \(i\in N\) a specific conjecture about what every other agent will play (i.e., a probability distribution defined on every \(A_{j}\) for \(j\ne i\)). More precisely, \( \mathbf {p}_{u}(\hat{a})\) postulates that each agent \(i\in N\) believes that any other player \(j\in N_{-i}\) will play action \(\hat{a}_{j}\) with probability \(p_{i}(\hat{a})\) and any other action \(a_{j}\ne \hat{a}_{j}\) with probability \(\frac{(1-p_{i}(\hat{a}))}{k_{j}-1}\).

We then say that the action profile \(\hat{a}\) is \(\mathbf {p}_{u}\)-dominant for \(\mathbf {p}_{u}(\hat{a})=\left( p_{1}(\hat{a}),\ldots ,p_{n}(\hat{a})\right) \) if, for any player \(i\in N\) and any \(j\in N_{-i}\), action \(\hat{a}_{i}\) is a best response to any probability distribution \(\lambda \left( \hat{a} \right) \in \Delta \left( A_{-i}\right) \) that assigns at least probability \( p_{i}(\hat{a})\) to the event of \(j\) playing action \(\hat{a}_{j}\) and lets \(j\) uniformly to randomize with the remaining probability over his alternative actions.

**Definition 1**

Some basic concepts in game theory can be formulated in terms of \(\mathbf {p} _{u}\)-dominance. For instance, an equilibrium in dominant strategies is \( \mathbf {p}_{u}\)-dominant with \(\mathbf {p}_{u}(a^{*})=\left( 0,\ldots ,0\right) \). In contrast, there is no vector \(\mathbf {p}_{u}\) for which a profile that contains dominated actions can turn out to be \(\mathbf {p}_{u}\)-dominant.

Focusing on Nash equilibria in pure strategies, every equilibrium is \( \mathbf {p}_{u}\)-dominant with \(\mathbf {p}_{u}(a^{*})=\left( 1,\ldots ,1\right) \): notice in fact that with such a (degenerate) vector of probability distributions, the definition of \(\mathbf {p}_{u}\)-dominance boils down to the requirement that every action in the profile under scrutiny is a best response to the actions taken by the other players, i.e., the definition of a Nash equilibrium. However, an equilibrium \(a^{*}\) is in general \(\mathbf {p}_{u}\)-dominant also for some other vectors \(\mathbf {p} _{u}(a^{*})\le \left( 1,\ldots ,1\right) \). More precisely, if the equilibrium \(a^{*}\) is \(\mathbf {p}_{u}\)-dominant with \(\mathbf {p} _{u}(a^{*}),\) then \(a^{*}\) is also \(\mathbf {p}_{u}\)-dominant for any \( \mathbf {p}_{u}^{\prime }(a^{*})\ge \mathbf {p}_{u}(a^{*})\).

As an example, consider the following \(3 \times 3\) coordination game where the action space of player \(i\in \left\{ A,B,C\right\} \) is given by \( A_{i}=\left\{ H,M,L\right\} \). Player \(A\) chooses the row, player \(B\) chooses the column, and player \(C\) chooses the matrix. In each cell payoffs appear in the order \(u_{A},u_{B}\) and \(u_{C}\).

\({a}_{B}{=H}\) | \({a}_{B}{=M}\) | \({a}_{B} {=L}\) | |
---|---|---|---|

\({a}_{A} =\,\, \)\(H\) | \({3,3,3}\) | \({2,0,2}\) | \({0,1,2}\) |

\({a}_{A} =\,\, \)\(M\) | \({0,2,2}\) | \({0,0,2}\) | \({2,3,1}\) |

\({a}_{A} =\,\, \)\(L\) | \({1,2,0}\) | \({3,2,1}\) | \({1,1,1}\) |

\({a}_{B}{=H}\) | \({a}_{B}{=M}\) | \({a}_{B} {=L}\) | |
---|---|---|---|

\({a}_{A} = \,\,\)\(H\) | \({2,2,0}\) | \({2,0,0} \) | \({1,3,2}\) |

\({a}_{A} =\,\,\)\(M\) | \({0,2,0}\) | \({3,3,3} \) | \({2,1,2}\) |

\({a}_{A} =\,\, \)\(L\) | \({3,1,2}\) | \({1,2,2} \) | \({0,0,1}\) |

\({a}_{B}{=H}\) | \({a}_{B}{=M}\) | \({a}_{B} {=L}\) | |
---|---|---|---|

\({a}_{A} =\,\,\)\(H\) | \({2,0,1}\) | \({1,2,3} \) | \({1,1,1}\) |

\({a}_{A} =\,\,\)\(M\) | \({2,1,3}\) | \({2,2,1} \) | \({1,0,0}\) |

\({a}_{A} =\,\,\)\(L\) | \({1,1,1}\) | \({0,1,0}\) | \({3,3,3}\) |

The game has three Nash equilibria: \(a^{*1}=(H,H,H)\), \(a^{*2}=(M,M,M) \), and \(a^{*3}=(L,L,L)\). These equilibria are Pareto equivalent such that the Pareto dominance criterion (i.e., the criterion that selects the Pareto superior equilibrium with the argument that this is the outcome upon which agents’ expectations should converge) does not refine the set \(A^{*}\).

The \(\mathbf {p}_{u}\)-dominance criterion selects instead a specific equilibrium. To find it, one needs to compute for every equilibrium \(a^{*\psi }\) with \(\psi \in \{ 1,2,3\} \) the functions \( E(a_{i}^{*\psi }|\lambda ( a^{*\psi }) )\), i.e., the expected payoff of action \(a_{i}^{*\psi }\) conditional on player \(i\)’s conjecture \(\lambda ( a^{*\psi }) \) that postulates that each opponent \(j\in N_{-i}\) plays action \(a_{j}^{*\psi }\) with probability \( \lambda (a_{j}^{*\psi }) =p_{i}\) and each of his alternative actions \(a_{j}\ne a_{j}^{*\psi }\) with probability \(\lambda ( a_{j}) =\frac{1-p_{i}}{2}\). Then, by imposing the conditions \( E(a_{i}^{*\psi }|\lambda ( a^{*\psi }) )\ge E(a_{i}|\lambda ( a^{*\psi }) )\) for any \(a_{i}\ne a_{i}^{*\psi }\), we find the components of the vector \(\bar{\mathbf {p}} _{u}\) which is the smallest vector for which the equilibrium \(a^{*\psi }\) is \(\mathbf {p}_{u}\)-dominant. Finally, we apply the \(\mathbf {p}_{u}\)-dominance criterion and select the equilibrium characterized by the smallest \(\bar{\mathbf {p}}_{u}\).

For instance, starting from the equilibrium \(a^{*1}\) and focusing without loss of generality on player \(A\) (the game is symmetric), we have the following: \(E(H|\lambda ( a^{*1}) )=\frac{5}{4}p_{i}^{2}+ \frac{1}{2}p_{i}+\frac{5}{4}\), \(E(M|\lambda ( a^{*1}) )=-2p_{i}+2\), and \(E(L|\lambda ( a^{*1}) )=-2p_{i}^{2}+2p_{i}+1\). Therefore, the equilibrium action \(H\) dominates action \({M}\) for any \(p_{i}\ge 0.265\) and action \({L}\) for any \(p_{i}\ge 0\). Given that similar relations also hold for players \(B\) and \(C\), \(a^{*1}\) is \(\mathbf {p}_{u}\)-dominant with \(\bar{\mathbf {p}} _{u}( a^{*1}) =( 0.265,0.265,0.265) \). Similar computations show that \(a^{*2}\) is \(\mathbf {p}_{u}\)-dominant with \( \bar{\mathbf {p}}_{u}( a^{*2}) =(0.547,0.547,0.547)\) while \( a^{*3}\) is \(\mathbf {p}_{u}\)-dominant with \(\bar{\mathbf {p}}_{u}( a^{*3}) =(0.771,0.771,0.771)\). The \(\mathbf {p}_{u}\)-dominance criterion thus unambiguously selects equilibrium \(a^{*1}=(H,H,H)\).

## 3 \(\mathbf {p}_{u}\)-dominance, risk-dominance and \(\mathbf {p}\)-dominance

In this section, we investigate the relationships that exist between the \( \mathbf {p}_{u}\)-dominance criterion and the risk-dominance (Harsanyi and Selten 1988) and \(\mathbf {p}\)-dominance (Morris et al. 1995; Kajii and Morris 1997) criteria. In the course of the analysis, we also explore how \( \mathbf {p}_{u}\)-dominance relates to mixed strategies equilibria.

**Proposition 1**

In any \(2 \times 2\) coordination game, the \(\mathbf {p}_{u}\)-dominance criterion always selects the risk-dominant equilibrium.

*Proof*

Consider a generic \(2 \times 2\) coordination game with \(i\in \left\{ A,B\right\} \) and \(A_{i}=\left\{ H,L\right\} \) with \(a>c\), \(d>b\), \(e>f\) and \(h>g\).

\({\small a}_{B}{\small \,=H}\) | \({\small a}_{B}{\small \,=L}\) | |
---|---|---|

\({\small a}_{A}{\small \,\,=\,\, H}\) | \(a,e\) | \(b,f\) |

\({\small a}_{A}{\small \,\,=\,\,L}\) | \(c,g\) | \(d,h\) |

The equilibrium \(a^{*1}=\left( H,H\right) \) is \(\mathbf {p}_{u}\)-dominant with \(\bar{\mathbf {p}}_{u}( a^{*1}) =\left( \frac{ d-b}{a-c+d-b},\frac{h-g}{e-f+h-g}\right) \), while \(a^{*2}=(L,L)\) is \( \mathbf {p}_{u}\)-dominant with \(\bar{\mathbf {p}}_{u}(a^{*2})=\left( \frac{ a-c}{a-c+d-b},\frac{e-f}{e-f+h-g}\right) \). Therefore, the \(\mathbf {p}_{u}\)-dominant criterion selects \(a^{*1}\) if \(\frac{d-b}{a-c+d-b}<\frac{a-c}{ a-c+d-b}\) and \(\frac{h-g}{e-f+h-g}<\frac{e-f}{e-f+h-g}\), i.e., if \(a-c>d-b\) and \(e-f>h-g\). But if both conditions are valid then \(\left( a-c\right) \left( e-f\right) >\left( d-b\right) \left( h-g\right) \), i.e., \(a^{*1}\) is the risk-dominant equilibrium because it is the equilibrium characterized by the highest product of the deviation losses. Similarly, if the \(\mathbf {p} _{u}\)-dominant criterion selects \(a^{*2}\), then it must be the case that \(\left( d-b\right) \left( h-g\right) >\left( a-c\right) \left( e-f\right) \) which means that \(a^{*2}\) is the risk-dominant equilibrium. \(\square \)

The proof of Proposition 1 provides an example of a relation that holds more generally in any \(2 \times 2\) game with two equilibria in pure strategies: if an equilibrium \(a^{*\psi }\in \left\{ a^{*1},a^{*2}\right\} \) is \( \mathbf {p}_{u}\)-dominant with \(\bar{\mathbf {p}}_{u}(a^{*\psi })=\left( \bar{p}_{1}(a^{*\psi }),\bar{p}_{2}(a^{*\psi })\right) , \) then \(\bar{p }_{i}(a^{*\psi })=q_{j}\) where \(q_{j}\) is the probability that defines the mixed-strategy equilibrium of the game: \(\left( q_{1}a_{1}^{*\psi }+\left( 1-q_{1}\right) a_{1}\text {, }q_{2}a_{2}^{*\psi }+\left( 1-q_{2}\right) a_{2}\right) \) with \(a_{i}\ne a_{i}^{*\psi }\). The intuition is the following: in a mixed-strategy equilibrium, agents randomize in such a way as to make the other player indifferent about what to play. This means that if player \(i\) attaches probability \(\bar{p} _{i}(a^{*\psi })=q_{j}\) to the event of \(j\) playing action \(a_{j}^{*\psi }\) [and thus probability \(\left( 1-\bar{p}_{i}(a^{*\psi })\right) =\left( 1-q_{j}\right) \) to the event of \(j\) playing \(a_{j}\) given that \( k_{j}-1=1\)], then both actions in \(A_{i}\) are best responses for player \(i\). However, for any \(q_{j}^{\prime }>q_{j}\), action \(a_{i}^{*\psi }\) becomes player \(i\)’s unique best response. It follows that the equilibrium \( a^{*\psi }\) is \(\mathbf {p}_{u}\)-dominant with \(\bar{\mathbf {p}} _{u}(a^{*\psi })=\left( \bar{p}_{1}(a^{*\psi }),\bar{p}_{2}(a^{*\psi })\right) \) where \(\bar{p}_{1}(a^{*\psi })=q_{2}\) and \(\bar{p} _{2}(a^{*\psi })=q_{1}\).

We now move to a comparison between the \(\mathbf {p}_{u}\)-dominance criterion and the \(\mathbf {p}\)-dominance criterion. As defined in Morris et al. (1995) for the case with two players and then extended by Kajii and Morris (1997) to the many players case, an equilibrium \(a^{*}\) is \(\mathbf {p}\)-dominant with \(\mathbf {p}(a^{*})=( p_{1}(a^{*}),\ldots ,p_{n}(a^{*})) \) if, for any agent \(i\in N\), action \( a_{i}^{*}\) is a best response to any probability distribution \(\lambda (a^{*})\in \Delta ( A_{-i}) \) such that \(\lambda ( a_{j}^{*}) \ge p_{i}(a^{*})\) for any \(j\ne i\).^{2} In other words, action \( a_{i}^{*}\) is \(\mathbf {p}\)-dominant if it maximizes player \(i\)’s expected payoff whenever \(i\) thinks that each one of the other players will play with probability not smaller than \(p_{i}(a^{*}),\) his component of the equilibrium profile. The difference with respect to \(\mathbf {p}_{u}\)-dominance is that \(\mathbf {p}\)-dominance does not require the remaining probability \(( 1-\lambda ( a_{j}^{*})) \) to follow any particular distribution over the alternative actions \(a_{j}\ne a_{j}^{*}\). While at first sight \(\mathbf {p}\)-dominance may seem a more general concept with respect to \(\mathbf {p}_{u}\)-dominance, notice that every Nash equilibrium is always simultaneously \(\mathbf {p}\)-dominant for some \(\mathbf {p}\) as well as \(\mathbf {p}_{u}\)-dominant for some \(\mathbf {p} _{u}\). What may differ across the two concepts are the smallest probability vectors \(\bar{\mathbf {p}}\left( a^{*}\right) \) and \(\bar{\mathbf {p}} _{u}\left( a^{*}\right) \) for which the equilibrium is dominant.

**Proposition 2**

Let \(a^{*}\) be a pure Nash equilibrium of the game \(G\). Then \(a^{*}\) is \(\mathbf {p}\)-dominant with \(\bar{\mathbf {p}}\left( a^{*}\right) \) as well as \(\mathbf {p}_{u}\)-dominant with \(\bar{\mathbf {p}}_{u}\left( a^{*}\right) \) and the relation \(\bar{\mathbf {p}}_{u}\left( a^{*}\right) \le \bar{\mathbf {p}}\left( a^{*}\right) \) holds.

*Proof*

In the appendix. \(\square \)

Proposition 2 clarifies the relationship that exists between the \(\bar{\mathbf {p}}\left( a^{*}\right) \) and \(\bar{\mathbf {p}}_{u}\left( a^{*}\right) \) vectors that characterize any Nash equilibrium. However, one thing is the definition and the computation of these vectors: a different issue is how \(\mathbf {p}\)-dominance and \(\mathbf {p}_{u}\)-dominance perform in terms of equilibrium selection. The two criteria in fact do not always lead to the same solution. More precisely, there exist games in which the \( \mathbf {p}_{u}\)-dominance criterion discriminates among equilibria that on the contrary the \(\mathbf {p}\)-dominance criterion deems equivalent (see Example 1 below) as well as games in which the opposite holds true (see Example 2 later).

*Example 1*

Consider the following game:

\({a_{B}}=\)\(H\) | \({a_{B}}{=M}\) | \({a}_{B} {=L}\) | |
---|---|---|---|

\({a_{A}}=\)\(H\) | 4,4 | 2,3 | 0,0 |

\({{a}_{A}}=\)\(M\) | 3,2 | 0,0 | 2,0 |

\({a_{A}}=\)\(L\) | 0,0 | 0,2 | 4,4 |

The game has two Nash equilibria: \(a^{*1}=(H,H)\) and \(a^{*2}=(L,L)\). Both equilibria are \(\mathbf {p}\)-dominant with \(\bar{\mathbf {p}} \left( a^{*1}\right) =\bar{\mathbf {p}}\left( a^{*2}\right) \mathbf {=} \left( \frac{1}{2},\frac{1}{2}\right) \). The \(\mathbf {p}\)-dominant criterion thus does not refine the set of equilibria. On the contrary, \(a^{*1}\) is \(\mathbf {p}_{u}\)-dominant with \(\bar{\mathbf {p}}_{u}\left( a^{*1}\right) \mathbf {=}\left( \frac{1}{5},\frac{1}{5}\right) \) while \(a^{*2}\) is \( \mathbf {p}_{u}\)-dominant with \(\bar{\mathbf {p}}_{u}\left( a^{*1}\right) \mathbf {=}\left( \frac{3}{7},\frac{3}{7}\right) \). Therefore, the \(\mathbf {p} _{u}\)-dominance criterion unambiguously selects the equilibrium \(a^{*1}=(H,H)\).

In the game of Example 1, equilibrium selection through \(\mathbf {p}_{u}\)-dominance is thus more effective with respect to a selection based on \( \mathbf {p}\)-dominance. And indeed, given the overall structure of the game, the selection of \(a^{*1}=(H,H)\) seems a sensible prediction: \(a^{*1}\) and \(a^{*2}\) are in fact Pareto equivalent but action \(a_{i}=H\) is more rewarding/less risky.

More in general the intuition as to why in some games the \(\mathbf {p}_{u}\)-dominance criterion manages to discriminate among equilibria that the \( \mathbf {p}\)-dominance criterion judges as equal is the following. The \( \mathbf {p}_{u}\)-dominance conjecture assigns to every player a belief system that attaches positive probability to the occurrence of any action that belongs to the action space of the opponents. On the contrary, the \(\mathbf {p }\)-dominance conjecture does not. The \(\mathbf {p}\)-dominance conjecture stems in fact from a worst-case scenario analysis that requires the equilibrium action under scrutiny (\(a_{i}^{*}\)) to dominate any alternative action (\(a_{i}\ne a_{i}^{*}\)) even under the most unfavorable probability distribution. This distribution is the one that allocates all the available probability \((1-\bar{p}_{i}(a^{*}))\) to the occurrence of the action(s) \(\tilde{a}_{j}\) for any \(j\ne i\) where \(\tilde{a }_{j}\) is the action that supports the profile for which the payoff difference \(u_{i}(a_{i},\tilde{a}_{-i})-u_{i}(a_{i}^{*},\tilde{a}_{-i})\) is maximal. As such the \(\mathbf {p}\)-dominance conjecture attaches zero probability to the event that an opponent plays action \(a_{j}\ne a_{j}^{\prime }\) with \(a_{j}^{\prime }\in \{ a_{j}^{*},\tilde{a} _{j}\} \). The conjecture thus disregards some of the opponents’ actions and therefore it ignores the associated payoffs.^{3} In other words, \(\mathbf {p}\)-dominance captures less information with respect to \(\mathbf {p}_{u}\)-dominance. However, while in general this is a positive feature of \(\mathbf {p}_{u}\)-dominance as it means that the \(\mathbf {p}_{u}\)-dominance criterion better captures the inherent structure of the strategic situation under study, the consequences of this fact in terms of equilibrium selection can go in both directions. In fact, and as it has already been mentioned, there also exist games in which the \(\mathbf {p}\)-dominance criterion selects a unique equilibrium while the \( \mathbf {p}_{u}\)-dominance criterion does not.

*Example 2*

\({a}_{B}{=H}\) | \({a}_{B}{=M}\) | \({a}_{B} {=L}\) | |
---|---|---|---|

\({a}_{A} = \,\,\)\(H\) | 4,4 | 2,2 | 0,0 |

\({a}_{A} =\,\,\)\(M\) | 2,2 | 0,0 | 1,3 |

\({a}_{A} =\,\,\)\(L\) | 0,0 | 3,1 | 3,3 |

The game has two Nash equilibria: \(a^{*1}=(H,H)\) and \(a^{*2}=(L,L)\). The \(a^{*1}\) equilibrium is \(\mathbf {p}\)-dominant with \( \bar{\mathbf {p}}\left( a^{*1}\right) \mathbf {=}\left( \frac{3}{7},\frac{3 }{7}\right) \) while the \(a^{*2}\) equilibrium is \(\mathbf {p}\)-dominant with \(\bar{\mathbf {p}}\left( a^{*2}\right) \mathbf {=}\left( \frac{4}{7}, \frac{4}{7}\right) \). The \(\mathbf {p}\)-dominant criterion thus selects the equilibrium \(a^{*1}=(H,H)\). Both equilibria are also \(\mathbf {p}_{u}\) -dominant with \(\bar{\mathbf {p}}_{u}\left( a^{*1}\right) =\bar{\mathbf {p} }_{u}\left( a^{*2}\right) \mathbf {=}\left( \frac{1}{3},\frac{1}{3} \right) \) such that in this case the \(\mathbf {p}_{u}\)-dominance criterion does refine the set of equilibria.

In what follows we focus on symmetric games and provide a sufficient condition that ensures that the \(\mathbf {p}_{u}\)-dominance criterion selects a unique equilibrium. Therefore, the same condition also guarantees that equilibrium selection through \(\mathbf {p}_{u}\)-dominance is weakly finer than a selection through \(\mathbf {p}\)-dominance. As an intermediary result, Lemma 1 shows that whenever the \(\mathbf {p}_{u}\)-dominance criterion selects multiple equilibria then it must be the case that all the selected equilibria are \(\mathbf {p}_{u}\)-dominant with \(\bar{\mathbf {p}}_{u}=\left( \frac{1}{k},\ldots ,\frac{1}{k}\right) \) where \(k=\left| A_{j}\right| \) for any \(j\in N\) is the number of actions at player \(j\)’s disposal. Proposition 3 then specifies the above-mentioned sufficient condition.

**Lemma 1**

Let \(A^{*}=\left\{ a^{*1},\ldots ,a^{*\Psi }\right\} \) with \(\Psi \ge 2\) be the set of Nash equilibria of a symmetric game \(G\) with \(k\) actions for each player. If the \(\mathbf {p}_{u}\)-dominance criterion selects the set of equilibria \(A^{**}=\left\{ a^{**1},\ldots ,a^{**\Phi }\right\} \) with \(\Phi \ge 2\) and \(A^{**}\subseteq A^{*}\) then \(\bar{\mathbf {p}}_{u}(a^{**\phi })=\left( \frac{1}{k} ,\ldots ,\frac{1}{k}\right) \) for any \(a^{**\phi }\in A^{**}\).

*Proof*

In the appendix. \(\square \)

**Proposition 3**

Let \(A^{*}=\left\{ a^{*1},\ldots ,a^{*\Psi }\right\} \) with \(\Psi \ge 2\) be the set of Nash equilibria of a symmetric game \(G\) with \(k\) actions for each player. A sufficient condition for the \(\mathbf {p}_{u}\)-dominance criterion to select a unique equilibrium \(a^{**}\in A^{*}\) is that \(\sum \nolimits _{a_{-i}\in A_{-i}}u_{i}(a_{i}^{*\psi },a_{-i})\ne \sum \nolimits _{a_{-i}\in A_{-i}}u_{i}(a_{i}^{*\xi },a_{-i})\) for any \(a_{i}^{*\psi }\) and \(a_{i}^{*\xi }\ne a_{i}^{*\psi }\).

*Proof*

In the appendix. \(\square \)

## 4 Conclusions

This paper introduced the concept of \(\mathbf {p}_{u}\)-dominance and proposed the \(\mathbf {p}_{u}\)-dominance criterion as a tool to refine multiple equilibria in normal form games. The criterion selects the equilibrium (or the set of equilibria) whose supporting actions are \(\mathbf {p}_{u}\)-dominant under the less stringent set of beliefs. The intuition for such a choice is that the selected equilibrium is the one that better solves the trade-off between the profitability of the equilibrium outcome and the riskiness of playing the supporting action in case some of the opponents deviate. The paper also explored the tight connections that exist between \( \mathbf {p}_{u}\)-dominance and \(\mathbf {p}\)-dominance (Morris et al. 1995; Kajii and Morris 1997). The two concepts are clearly similar in their premises as well as in their scope but we feel \(\mathbf {p}_{u}\)-dominance presents some advantages. First, \(\mathbf {p}_{u}\)-dominance relies on a conjecture that is more in line with the actual behavior of a relevant fraction inexperienced players (so called “wordly” archetype). Second, it is more easily computable. In particular, one does not need to identify the most unfavorable distribution that defines \(\mathbf {p}\)-dominance, a task that may turn out to be not so trivial especially in games that involve more than two players and/or actions. Finally, \(\mathbf {p}_{u}\)-dominance more fully captures the inherent structure of the game under study as it takes into account any possible deviation of the other players.

## Footnotes

- 1.
The fact that the conjecture posits that players use a uniform distribution to randomize over the actions that do not belong to the profile under scrutiny explains the subscript “\(u\)” in the term “\(\mathbf {p}_{u}\)-dominance”. This should not be confused with the concept of \(u\)-dominance Kojima (2006) that has been proposed as a criterion for equilibrium selection based on perfect foresight dynamics and that assumes that the number of opponents adopting a certain strategy follows a uniform distribution.

- 2.
Tercieux (2006a, b) further extends the concept of \(\mathbf {p}\)-dominance by introducing the notion of \(\mathbf {p}\)-best response set: a set profile \(S=(S_{1},\ldots ,S_{n})\) is a \(\mathbf {p}\)-best response set if for every \(i\) the set \(S_{i}\) contains an action that best responds to any conjecture that assigns probability of at least \(p_{i}\) to the event that other players select their action from \(S_{-i}\).

- 3.
Notice that the actions that the \(\mathbf {p}\)-dominance conjecture disregards are not necessarily dominated (see for instance action \(M\) in the game in Example 1). Notice also that the \(\mathbf {p}_{u}\)-dominance conjecture suffers instead from the opposite problem as it assigns positive probability to the event that the opponents play strictly dominated actions (if these exist). Still, this shortcoming can be easily fixed by applying the \(\mathbf {p}_{u}\)-dominance criterion only after the elimination of all strictly dominated actions.

- 4.
If an undominated action does not exist then \(a^{*}\) is \(\mathbf {p}_{u}\)-dominant as well \(\mathbf {p}\)-dominant with \(\bar{\mathbf {p}}_{u}\left( a^{*}\right) =\bar{\mathbf {p}}\left( a^{*}\right) =(0,\ldots ,0)\).

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