# strategy gambling money game

Two players, A and B, are each given 10000 dollars. They'll play a 10 round game. In each round both have to gamble some of this money (can be zero); in the 10th round, they both must use all their remaining money. The winner of each round is the one who gambles less. He gets both sums gambled in that round; these winnings are put away and not added to the money available for gambling. The winner of the game is the one who won more money.

What is the best strategy to win this kind of game?

Example for a 3 round-game: round 1, A gambles 5000, B gambles 4000 --> B gets 9000. Round 2, A 5000, B 4000 --> B gets 9000. Round 3, A 0, B 2000 --> A gets 2000, B wins getting 18000.

This seems like a zero-sum game theory, but there's something different, any help will be appreciated.

• You may be interested in this: en.wikipedia.org/wiki/Blotto_games, though it's not quite the same. – joriki Dec 13 '11 at 16:19
• This is also related: math.stackexchange.com/questions/76095/…. I thought about that one for quite a while and found that the analysis quickly becomes surprisingly complicated if the game has more than two rounds; I believe the same will be the case here. – joriki Dec 13 '11 at 16:24
• Note also that, from a game-theoretric viewpoint (i.e. ignoring issues of psychology and cognitive limitations), the question to be asked about a symmetric game like this is not what's the best strategy to win, since there can't be one, but what is the optimal stragegy in the sense of a Nash equilibrium. Note also that the existence of a Nash equilibrium is only guaranteed for games with a finite strategy set; there are examples of continuous games like this one without a Nash equilibrium. – joriki Dec 14 '11 at 10:25
• @joriki: like one in the prisoner's dilema, you mean? – zfm Dec 14 '11 at 11:08
• No. You should probably read up on the prisoner's dilemma and/or on Nash equilibria if you thought that. I couldn't have meant the prisoner's dilemma both because it has a finite strategy set and because it has a straightforward (pure) Nash equilibrium, in which both players always defect. I saw an example a while ago; I'll post a link if I find it again. – joriki Dec 14 '11 at 13:42