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So we have a game in which 2 players, P1 and P2, are randomly given a number $x_1,x_2\in [0,1]$. Each player first antes \$1, are given their numbers, and then P1 can choose to bet any value $B$, or pass (in which case, who ever had the higher number wins the pot). If P1 bets, P2 can then either call the bet, or fold.

We are asked to find an optimal strategy for P1 and P2 (which is dependent on $B$). Any tips how to get started with this one?

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    $\begingroup$ This is really more of a poker type game, where playing the person is far superior to any mathematical strategy. $\endgroup$ – Jonny Sep 9 '14 at 1:56
  • $\begingroup$ By fold, do you mean that P1 will get $\$B+2$? Surely, in that case P2 will never fold, and 'passing' can be treated as a special case of $B=0$. $\endgroup$ – theindigamer Sep 9 '14 at 2:00
  • $\begingroup$ No... folding means that P1 gets a gain of +1, as P2 did not call the bet, and P1 had put one dollar in initially. $\endgroup$ – user3784030 Sep 9 '14 at 2:05
  • $\begingroup$ I think the other commenters are right. P2 has no information other than the fact that he loses a dollar if he folds, so he might as well call every time. I don't know what an optimal strategy for P1 dependent on B would be when B depends on P1. $\endgroup$ – NoName Sep 9 '14 at 2:16
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    $\begingroup$ This is von Neumann's poker model. See, for example, Chris Ferguson and Thomas S. Ferguson, "On the Borel and von Neumann poker models", Game Theory and Appl. 9 (2003), 17-32; a pdf is available on Tom Ferguson's web page. $\endgroup$ – bof Sep 9 '14 at 2:56

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