<p>It will be useful to spell out the relevant definitions.</p>

<p>Let $A_i$ be the set of possible actions for player $i$, $A_{-i}$ be the set of possible joint actions of all players except for player $i$, $A$ be the set of possible joint actions of all the players. For $a=(a_i,a_{-i})\in A$, let $u_i(a)$ denote the payoff to player $i$ from action $a_i$, given that the other players play $a_{-i}$.</p>

<p>Definition 1: An action $a_i\in A_i$ is weakly dominant for player $i$ if for every $a_{-i}\in A_{-i}$, $u_i(a_i,a_{-i})\ge u_i(\overline{a_i},a_{-i})$ for every $\overline{a_i}\in A_i$, i.e. no matter what the other players do, action $a_i$ yields a payoff at least as high as any other action available to player $i$. [Strict dominance given by $&gt;$ instead of $\ge$.]</p>

<p>Definition 2: An action $a_i\in A_i$ is a best response to action $a_{-i}\in A_{-i}$ if $u_i(a_i,a_{-i})\ge u_i(\overline{a_i},a_{-i})$ for every $\overline{a_i}\in A_i$, i.e. If I fix other players' actions to some particular joint action, then $a_i$ yields a payoff to player $i$ at least as high as any other action available to player $i$.</p>

<p>Claim 1: If an action $a_i\in A_i$ is strictly or weakly dominant, then it is a best response for any joint action $a_{-i}\in A_{-i}$. [This follows from Definitions 1 and 2 immediately.]</p>

<p>Definition 3: A joint action $a\in A$ is a Nash equilibrium if for each player $i$, action $a_i$ is a best response to $a_{-i}$.</p>

<p>Claim 2 (your question): A strategy profile (i.e. joint action) $s=(s_1,...,s_n)\in A$ in which each $s_i$ is a (strictly or weakly) dominant strategy is a Nash equilibrium.</p>

<p>Note that Claim 2 is a consequence of Claim 1.</p>

CC BY-SA 3.0
2016-02-22T03:09:55.687Z
2016-02-22T03:09:55.687Z
manofbear
<p>I am new to game theory and I came across this line, " A strategy profile (s1, . . . , sn) in which every si is dominant for agent i (strictly, weakly, or very weakly) is a Nash equilibrium."</p>

<p>But why is that? And how would an equilibrium form in strictly dominant strategies? Do they both just yield the same best payoff?</p>

<p>Any help would be appreciated!</p>

<game-theory><nash-equilibrium>
Show that an equilibrium in strictly dominant strategies is a unique Nash equilibrium
bZhang
# Show that an equilibrium in strictly dominant strategies is a unique Nash equilibrium

(Asked by: username_0 on 2016-02-21)

I am new to game theory and I came across this line, " A strategy profile (s1, . . . , sn) in which every si is dominant for agent i (strictly, weakly, or very weakly) is a Nash equilibrium."

But why is that? And how would an equilibrium form in strictly dominant strategies? Do they both just yield the same best payoff?

Any help would be appreciated!

**Tags:** game-theory, nash-equilibrium


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## 1 answers

### Answer by username_1 on 2016-02-22. Score: 5 | Accepted answer

It will be useful to spell out the relevant definitions.

Let $A\_i$ be the set of possible actions for player $i$, $A\_{-i}$ be the set of possible joint actions of all players except for player $i$, $A$ be the set of possible joint actions of all the players. For $a=(a\_i,a\_{-i})\in A$, let $u\_i(a)$ denote the payoff to player $i$ from action $a\_i$, given that the other players play $a\_{-i}$.

Definition 1: An action $a\_i\in A\_i$ is weakly dominant for player $i$ if for every $a\_{-i}\in A\_{-i}$, $u\_i(a\_i,a\_{-i})\ge u\_i(\overline{a\_i},a\_{-i})$ for every $\overline{a\_i}\in A\_i$, i.e. no matter what the other players do, action $a\_i$ yields a payoff at least as high as any other action available to player $i$. [Strict dominance given by $>$ instead of $\ge$.]

Definition 2: An action $a\_i\in A\_i$ is a best response to action $a\_{-i}\in A\_{-i}$ if $u\_i(a\_i,a\_{-i})\ge u\_i(\overline{a\_i},a\_{-i})$ for every $\overline{a\_i}\in A\_i$, i.e. If I fix other players' actions to some particular joint action, then $a\_i$ yields a payoff to player $i$ at least as high as any other action available to player $i$.

Claim 1: If an action $a\_i\in A\_i$ is strictly or weakly dominant, then it is a best response for any joint action $a\_{-i}\in A\_{-i}$. [This follows from Definitions 1 and 2 immediately.]

Definition 3: A joint action $a\in A$ is a Nash equilibrium if for each player $i$, action $a\_i$ is a best response to $a\_{-i}$.

Claim 2 (your question): A strategy profile (i.e. joint action) $s=(s\_1,...,s\_n)\in A$ in which each $s\_i$ is a (strictly or weakly) dominant strategy is a Nash equilibrium.

Note that Claim 2 is a consequence of Claim 1.


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