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Consider a two player simultaneous game where each player chooses a non-negative value, $x_i$, $i=1,2$. The payoff to each player is $π_i(x_1,x_2)=2x_i+2x_1x_2-x_i^2.$

a. What are the pure strategy Nash equilibria of this game?

b. Suppose that the set of strategies is limited to $[0,M],M>0$. What are the pure strategy Nash equilibria of the game now?

For a) I got the answer that there are no PSNE since there's no intersection between the best response function of both players. Please correct me if that's wrong!

For b) I got very confused because I'm unsure how bounding the values of $x$ will make the conclusion any different from a). I realize that the payoff function provided doesn't necessarily mean that the larger $x_i$ is, the larger the payoff will be.

Below is my proof:

Consider $π_i(x_1,x_2)=2x_i+2x_1x_2-x_i^2.$. We first assume $x_1> x_2$. In order to find out if a larger number leads to a higher payoff, i.e.

if $2x_1+2x_1x_2-x_1^2>2x_2+2x_1x_2-x_2^2$

$2x_1-x_1^2>2x_2-x_2^2$

$x_1(2-x_1)> x_ 2(2-x_2)$

Following from $x_1>x_2$,

$2-x_1<2-x_2$

$x_1(2-x_1)> x_ 2(2-x_2)$ for $0<x_1\leq 1 $ and $0<x_2< 1$

and

$x_2(2-x_2)> x_ 1(2-x_1)$ for $x_1>1 $ and $x_2\ge 1$, where $x_2<x_1$

Therefore, in order to maximize the payoff of each player and find the intersection of their best response such that they will have no incentive to deviate, with the condition that $[0,M]$, we find that:

As long as $0<M\leq 1$, the PSNE will be {$P1: M, P2: M$}. When $M>1$, the PSNE will be {$P1: 1, P2: 1$}.

I'm quite unsure if I'm right though, and would love to get some input!

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    $\begingroup$ So if $M = 2$ then $x_1 = x_2 = 1$ is a Nash-equilibrium? The constraints are not active here, meaning this is also a Nash-equilibrium of the original game. But that game had none! $\endgroup$ – Giskard Nov 5 '17 at 13:56
  • $\begingroup$ @denesp oh that makes sense! But why did my working indicate that a higher $x_i$ value doesn't necessarily indicate a higher payoff? Wouldn't that mean that both players aren't maximizing their payoff if the only nash equilibrium is at $M,M$ regardless of the range? $\endgroup$ – iamtrying Nov 5 '17 at 14:41
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    $\begingroup$ Comparing the two players' payoffs just does not mean anything. Let's play a game. Whoever says the bigger number wins a dollar from the other. If we both say the same number, it is a draw. Now if we both say five, it is a draw. We both get a payoff of 0. Neither payoff is larger than the other. Yet both of us saying five is obviously not an equilibrium. $\endgroup$ – Giskard Nov 5 '17 at 14:46
  • $\begingroup$ I recommend reading about constrained optimization. That seems to be your problem here, not game theory. $\endgroup$ – Giskard Nov 5 '17 at 14:47
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    $\begingroup$ @HerrK. I would rather vote to close. The problem is homework level and very specific. $\endgroup$ – Giskard Nov 8 '17 at 21:16

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