<p>The following proposition is well known: </p>

<blockquote>
  <p>If a stage game <span class="math-container">$G$</span> has a unique Nash equilibrium, then for any finite <span class="math-container">$T$</span>, the repeated game <span class="math-container">$G(T)$</span> has a unique subgame perfect equilibrium outcome in which the Nash equilibrium of <span class="math-container">$G$</span> is played in every stage.</p>
</blockquote>

<p>Since your stage game has a unique NE of <span class="math-container">$(T,R)$</span>, this must be the outcome of the SPE of any finitely repeated game based on this stage game. Specifically regarding your question, it is impossible for <span class="math-container">$(B,L)$</span> to be played (in any stage including the first) in any SPE of the <span class="math-container">$T$</span>-times repeated game.</p>

CC BY-SA 4.0
2020-01-04T04:32:31.153Z
2020-01-04T04:32:31.153Z
Herr K.
<p><strong>Existence of a subgame perfect Nash-equilibrium</strong></p>

<p>Given is the following game</p>

<p><a href="https://i.sstatic.net/cVxwe.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/cVxwe.png" alt="2 player game"></a></p>

<p>The game is repeated finitely many times and the total payoff is the sum of the payoff from each repetition.</p>

<p>If we assume that <span class="math-container">$T=2$</span>, is it then possible to find a subgame perfect Nash-equilibrium such that the strategy <span class="math-container">$(B,L)$</span> is played in the first stage (first repetition)?</p>

<p>So far as I can count, it shouldn't be possible to construct any threats for <span class="math-container">$T=2$</span> such that the <span class="math-container">$(B,L)$</span> strategy is played in the first stage. This is due to the Nash-equilibrium laying in <span class="math-container">$(T,R)$</span> and it takes <em>at least</em> <span class="math-container">$T\geq 3$</span> to ensure that <span class="math-container">$(B,L)$</span> is played in the first stage. </p>

<p>Though, I'm not quite sure how to prove my statement. Can any of you confirm/deny my thoughts and help me with some arguments? Thanks.</p>

<game-theory>
Existence of a subgame perfect Nash-equilibrium
mas2
# Existence of a subgame perfect Nash-equilibrium

(Asked by: username_0 on 2020-01-03)

**Existence of a subgame perfect Nash-equilibrium**

Given is the following game

[![2 player game](https://i.sstatic.net/cVxwe.png)](https://i.sstatic.net/cVxwe.png)

The game is repeated finitely many times and the total payoff is the sum of the payoff from each repetition.

If we assume that $T=2$, is it then possible to find a subgame perfect Nash-equilibrium such that the strategy $(B,L)$ is played in the first stage (first repetition)?

So far as I can count, it shouldn't be possible to construct any threats for $T=2$ such that the $(B,L)$ strategy is played in the first stage. This is due to the Nash-equilibrium laying in $(T,R)$ and it takes *at least* $T\geq 3$ to ensure that $(B,L)$ is played in the first stage.

Though, I'm not quite sure how to prove my statement. Can any of you confirm/deny my thoughts and help me with some arguments? Thanks.

**Tags:** game-theory

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## 1 answers

### Answer by username_1 on 2020-01-04. Score: 3 | Accepted answer

The following proposition is well known:

> If a stage game $G$ has a unique Nash equilibrium, then for any finite $T$, the repeated game $G(T)$ has a unique subgame perfect equilibrium outcome in which the Nash equilibrium of $G$ is played in every stage.

Since your stage game has a unique NE of $(T,R)$, this must be the outcome of the SPE of any finitely repeated game based on this stage game. Specifically regarding your question, it is impossible for $(B,L)$ to be played (in any stage including the first) in any SPE of the $T$-times repeated game.


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