Funny betting game
So, I play poker sometimes, and one aspect that really interests me is the game-theoretic aspects to betting. Here's an interesting abstraction of a certain situation involving repeated bets (i.e. pre-flop, flop, turn, river).
There are just two players. Each player can put any amount between $1 and $100 into the pot, and each chooses his amount secretly and independently of the other. If the amounts are unequal, then whoever put more money into the pot wins.
Now, there are two superpowers in this game, a tie-breaking superpower and a pot-stealing superpower. the tie-breaking superpower is you get to win in the event of a tie, which means if you put in $100, the other player can't beat you by putting in a bigger amount. The pot-stealing superpower is invoked when an ace comes off of the top of a shuffled deck of cards, in which case the player with that superpower automatically wins all the money. The pot-stealing superpower takes precedence over the tie-breaking superpower.
The question is: which superpower would you pick? Also, how would you play the game?
posted by Jeremy @ 1:22 PM
1 Comments:
You are definitely right that any consistent betting pattern is a massive disadvantage, especially to an adaptive opponent. The game-theoretically optimial randomized strategy (defined as the strategy with the highest expected value against that strategy's "most feared" counterstrategy) for each player involves making every possible bet sometimes. Of course, this dosen't mean making every bet equally often. In fact, both players have a strong preference for betting low amounts.
Because of the fact that all possible strategies have a positive probability, it turns out that you can write the optimial strategy using just plain old linear algebra. I'll leave this as an exercise for the reader :-)
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