<p>Yes, these two concepts are quite independent. Consider the following 2-player game: First player A will choose 'left' or 'right', then player B decides whether both players will 'get money' or 'lose money'.</p>
<p>The payoff of B is <span class="math-container">$1$</span> or <span class="math-container">$-1$</span> depending on their action. The payoff of A is <span class="math-container">$1$</span> if B decided both of them should 'get money', and in this case A's own action does not affect the payoffs. If B decided both of them should lose money, then A gets a payoff of <span class="math-container">$-1$</span> if they chose 'left' and <span class="math-container">$-2$</span> if they chose 'right'.</p>
<p>In this case choosing 'right' is weakly dominated, but that is quite alright, because B will choose 'get money' anyway. Thus ('right', ('get money','get money')) is a SPE.</p>
<hr />
<p>There are some equilibrium concepts, such as the <em>sequential equilibrium</em>, where <em>off the equilibrium path</em> players have to entertain the idea that any action will be taken with some positive probability. Such a refinement would eliminate the equilibria in the above example. As <a href="https://economics.stackexchange.com/questions/57078/can-a-subgame-perfect-equilbrium-consist-of-a-weakly-dominated-action/57080?noredirect=1#comment93599_57080">Michael Greinecker points out in his comment</a>, even this refinement does not always eliminate weakly dominated strategies.</p>

CC BY-SA 4.0
2023-12-02T18:20:51.003Z
In a simultaneous move game, every Nash equilibrium is a sequential equilibrium. So sequential equilibria can include weakly dominated actions.
Michael Greinecker
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2023-12-02T21:56:11.210Z
@MichaelGreinecker Indeed. I find most of these refinements are surprisingly brittle.
Giskard
CC BY-SA 4.0
2023-12-02T15:17:51.347Z
2023-12-02T21:58:36.330Z
2023-12-02T21:58:36.330Z
Giskard
<p>I was reviewing the Market Entry game and observed that 'being mean to the entrant' is weakly dominated by 'being nice to the entrant.' Coincidentally, 'being nice to the entrant' is also part of the Subgame Perfect Equilibrium for the game.</p>
<p>Since this is the only sequential game with weak domination that I have encountered so far, I am curious to know whether weakly dominated actions could also constitute a Subgame Perfect Equilibrium for some sequential game.</p>

CC BY-SA 4.0
2023-12-02T15:07:53.167Z
Hi! Note that there are many variants to the Market Entry game, so your question is not well-defined.
Giskard
<microeconomics><game-theory><equilibrium>
Can a Subgame Perfect Equilbrium consist of a weakly dominated action?
wes BOREland
# Can a Subgame Perfect Equilbrium consist of a weakly dominated action?

(Asked by: username_0 on 2023-12-02)

I was reviewing the Market Entry game and observed that 'being mean to the entrant' is weakly dominated by 'being nice to the entrant.' Coincidentally, 'being nice to the entrant' is also part of the Subgame Perfect Equilibrium for the game.

Since this is the only sequential game with weak domination that I have encountered so far, I am curious to know whether weakly dominated actions could also constitute a Subgame Perfect Equilibrium for some sequential game.

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

> **Comments:**
> **username_1**: Hi! Note that there are many variants to the Market Entry game, so your question is not well-defined.  

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## 1 answers

### Answer by username_1 on 2023-12-02. Score: 1 | Accepted answer

Yes, these two concepts are quite independent. Consider the following 2-player game: First player A will choose 'left' or 'right', then player B decides whether both players will 'get money' or 'lose money'.

The payoff of B is $1$ or $-1$ depending on their action. The payoff of A is $1$ if B decided both of them should 'get money', and in this case A's own action does not affect the payoffs. If B decided both of them should lose money, then A gets a payoff of $-1$ if they chose 'left' and $-2$ if they chose 'right'.

In this case choosing 'right' is weakly dominated, but that is quite alright, because B will choose 'get money' anyway. Thus ('right', ('get money','get money')) is a SPE.

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There are some equilibrium concepts, such as the *sequential equilibrium*, where *off the equilibrium path* players have to entertain the idea that any action will be taken with some positive probability. Such a refinement would eliminate the equilibria in the above example. As [username_2 points out in his comment](https://economics.stackexchange.com/questions/57078/can-a-subgame-perfect-equilbrium-consist-of-a-weakly-dominated-action/57080?noredirect=1#comment93599_57080), even this refinement does not always eliminate weakly dominated strategies.

> **Comments:**
> **username_2**: In a simultaneous move game, every Nash equilibrium is a sequential equilibrium. So sequential equilibria can include weakly dominated actions.  
> **username_1**: @MichaelGreinecker Indeed. I find most of these refinements are surprisingly brittle.  

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