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Consider the following two-player game. The players simultaneously draw one sample each from a continuous random variable X, which follows $Uniform\ [0, 100]$. After observing the value of her own sample, which is private information (that is, opponent does not observe it), players simultaneously and independently choose one of the following: $SWAP$, $RETAIN$. If both the players choose $SWAP$ then they exchange their initially drawn numbers. Otherwise, if at least one person chooses $RETAIN$, both of them retain their numbers. A player earns as many Rupees as the number she is holding at the end of the game. what is the probability that the players will exchange their initially drawn numbers?

My approach

this is a question from one of master's entrance exams, now i think that the question is incomplete because given the information in the question if the it is up to player to draw any number from the given interval of $ [0,100]$, why would any player choose any number less than $100$. all the players will choose to maximize their payoff and hence they must draw $100$. and hence the required probability should be equal to 0

But if there is a missing point from the question, for instance the players might draw from 101 balls numbered 0 to 100 in that case the question seems like Monty Hall Problem.

If i am wrong then can anyone please help me with this question?

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  • $\begingroup$ I'm confused about your question. The set-up says each player draw exactly 1 sample from U(0,100), but then in your approach you ask why the players wouldn't just draw the number 100. If the players are given a random number, how could they possibly choose to draw the number 100? How could they possibly draw 101 numbers? You also seem to imply they are drawing discrete numbers 0-100, when the set-up says a continuous U(0,100) random value. $\endgroup$ May 13 at 13:18

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How about defining some cutoff value $q_1$ below which player 1 will decide to swap and a cut-off value $q_2$ below which player 2 will decide to swap. Then compute the win probability as a function of $q_1$ and $q_2$ and see whether there is a Nash equilibrium.

situation sketch

When $q_2 < q_1$ the player 1 can improve their strategy by decreasing $q_1$ as this will increase the region of lottery outcomes where the player 1 is gonna win, at the cost of a reduction in the region where they swap and each has 50:50 chance.

When $q_2 = q_1$ the player 1 can not improve their strategy.

When $q_1 < q_2$ the player 1 can improve the strategy by increasing $q_1$. This will increase the swap region at the cost of the region where player 2 would win.

So what we have here is a Nash equilibrium in all the points on the line $q_1 = q_2$. If $q_1 = q_2$ then none of the players can improve their strategy. However these equilibria are not stable, a small change in the strategy by the one player changes the strategy of the other. There is also no dominant strategy. There is no strategy that works the best no matter what your opponent does.

why would any player choose any number less than 100

Say a player has the number 99 then of course the player would still prefer a larger number but the probability that the swapped number is an improvement is very low.

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