0-1 game -- Problems 4 and 5
October 24th, 2012 at 9:09:26 AM
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What happens if players are free to choose the size of their bet?
Problem 5: The same as problem 1, except for the amount bet.
You may want to begin with Problem 4, which has a fixed value of b before play begins, instead of letting X choose it after seeing x.
Find the optimal strategies and the EV as functions of b.
b = 0 is problem 1. The EV was 0.10 in favour of X.
This is not a question of risk aversion: players are maximising expected value.
So the game is NOT dependant on subjective matters like utility functions or wallet size. The optimal value of bet size is built in the game description.
Problem 5: The same as problem 1, except for the amount bet.
- Player X and Y each ante $1.
- Both are given an independant number uniformly distributed from 0 to 1. The higher number wins.
- Player X may bet $1+b or check. (b >= 0)
- If player X checks then Y checks. The bet is resolved.
- If player X bets then Y may call or fold.
You may want to begin with Problem 4, which has a fixed value of b before play begins, instead of letting X choose it after seeing x.
Find the optimal strategies and the EV as functions of b.
b = 0 is problem 1. The EV was 0.10 in favour of X.
You'll find that the probability of betting as well as the EV are diminishing when the amount rises. Actually, EV rises slightly first (up to 0.1111 at b = 1) then drops towards zero in the limit (Stakes are too high, of course. Fair game, but quite silly: never bet. )
This is not a question of risk aversion: players are maximising expected value.
So the game is NOT dependant on subjective matters like utility functions or wallet size. The optimal value of bet size is built in the game description.
Reperiet qui quaesiverit
