<p>In a finitely repeated game with a unique NE, the only SPNE is the repetition of the unique NE. The reason is that by backward induction the NE will be played in the last period and, hence, also in the penultimate period, and so on.</p>

<p>Here, there are multiple NE so that the different NE can be used to reward and punish previous behavior. Your idea of how to construct such a SPNE is correct. The reward payoff in the second (so the final) period is 2 and the punishment payoff is 0 for each player.
So we only have our SPNE if <span class="math-container">$4+2 \geq x+0 \iff  x\leq 6$</span>. Otherwise, the reward is not profitable enough (or the punishment NE not harsh enough).</p>

<p>If <span class="math-container">$x&lt;4$</span>, <span class="math-container">$(Q,Q)$</span> is a NE, and repeating this NE twice would be a SPNE.</p>

CC BY-SA 4.0
2019-04-08T09:04:38.653Z
Thank You for the answer. Does the fact that x needs to be strictly less than 6 ensure that punishment is 'harsh'?
S.Rana
CC BY-SA 4.0
2019-04-08T09:08:26.647Z
I apologize, it only has to be weakly larger. If $x=6$, a unilateral deviation from the strategy profile you outlined would still not be (strictly) profitable.
Bayesian
CC BY-SA 4.0
2019-04-08T08:50:16.070Z
2019-04-08T09:06:27.160Z
2019-04-08T09:06:27.160Z
Bayesian
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<p>I approached this question in this way:
<span class="math-container">$(P_1,P_2), (R_1,R_2), (S_1,S_2)$</span> are the Nash Equilibria of the Stage 1 game. For the given strategy to be sustained as SPNE, there should be no way unilateral deviations increase payoffs of the individuals. When <span class="math-container">$(Q_1,Q_2)$</span> is played in stage 1, the payoff at the end of stage 2 would be <span class="math-container">$4+2=6$</span> each.
Now if player 1 deviates to a higher payoff, by playing <span class="math-container">$P1$</span>, the total payoff he will get after two stages would be <span class="math-container">$x+0$</span>
Deviation would be futile if <span class="math-container">$x \leq 6$</span>. Also, it is given tha <span class="math-container">$x&gt;4$</span>
So, one value of <span class="math-container">$x$</span> which can sustain the above strategy as SPNE is <span class="math-container">$5$</span>.
Please correct me where I am wrong. Thanks!</p>

<game-theory><self-study><nash-equilibrium><dynamic-games><repeated-games>
Repeated Game SPNE
S.Rana
# Repeated Game SPNE

(Asked by: username_0 on 2019-04-08)

[![enter image description here](https://i.sstatic.net/gkNfY.jpg)](https://i.sstatic.net/gkNfY.jpg)

I approached this question in this way:
$(P\_1,P\_2), (R\_1,R\_2), (S\_1,S\_2)$ are the Nash Equilibria of the Stage 1 game. For the given strategy to be sustained as SPNE, there should be no way unilateral deviations increase payoffs of the individuals. When $(Q\_1,Q\_2)$ is played in stage 1, the payoff at the end of stage 2 would be $4+2=6$ each.
Now if player 1 deviates to a higher payoff, by playing $P1$, the total payoff he will get after two stages would be $x+0$
Deviation would be futile if $x \leq 6$. Also, it is given tha $x>4$
So, one value of $x$ which can sustain the above strategy as SPNE is $5$.
Please correct me where I am wrong. Thanks!

**Tags:** game-theory, self-study, nash-equilibrium, dynamic-games, repeated-games


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## 1 answers

### Answer by username_1 on 2019-04-08. Score: 1 | Accepted answer

In a finitely repeated game with a unique NE, the only SPNE is the repetition of the unique NE. The reason is that by backward induction the NE will be played in the last period and, hence, also in the penultimate period, and so on.

Here, there are multiple NE so that the different NE can be used to reward and punish previous behavior. Your idea of how to construct such a SPNE is correct. The reward payoff in the second (so the final) period is 2 and the punishment payoff is 0 for each player.
So we only have our SPNE if $4+2 \geq x+0 \iff x\leq 6$. Otherwise, the reward is not profitable enough (or the punishment NE not harsh enough).

If $x<4$, $(Q,Q)$ is a NE, and repeating this NE twice would be a SPNE.

> **Comments:**
> **username_0**: Thank You for the answer. Does the fact that x needs to be strictly less than 6 ensure that punishment is 'harsh'?  
> **username_1**: I apologize, it only has to be weakly larger. If $x=6$, a unilateral deviation from the strategy profile you outlined would still not be (strictly) profitable.  

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