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#1
09-12-2007, 03:44 PM
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Game Theory Problem Of The Week
For this week's game theory problem we will take a look at another situation .
There are two players who pick numbers from 1-100 without replacement . Each player posts a $1 ante but player one must always check even though he's first to act . Player two has the option of betting the pot or checking behind . Given this knowledge , what strategy must player two employ to maximize his EV ? We may make the assumption that player one and two are playing optimally aside from the stipulation placed on player one . 
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#7
09-13-2007, 12:16 AM
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Re: Game Theory Problem Of The Week
[ QUOTE ]
For this week's game theory problem we will take a look at another situation . There are two players who pick numbers from 1-100 without replacement . Each player posts a $1 ante but player one must always check even though he's first to act . Player two has the option of betting the pot or checking behind . Given this knowledge , what strategy must player two employ to maximize his EV ? We may make the assumption that player one and two are playing optimally aside from the stipulation placed on player one . [/ QUOTE ] Optimal is: P1 [1,56] Fold [57,100] Call P2 [1,11] Bet [12,78] Fold [79,100] Bet P1 EV = -1/9 P1 can't improve by making any changes against P2 P2 can't improve by making any changes against P1
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#8
09-13-2007, 01:06 AM
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Re: Game Theory Problem Of The Week
Here is the EV set up using the cash game definition of EV .
Player two bets with [x,1] and player one check calls with [(2x+1)/3,1] . Notice that the answer to this will be the same as in the discrete case . EV(P2)= 1*(1-x)*(2x+1)/3 + 3*(1-x)*(2-2x)/3*1/3 -3*(1-x)*(2-2x)/3*2/3 - (1-x)/3 There are 4 different product terms : The first is your EV|player 1 folds . The second is your EV|player 1 calls and you win The third is your EV|player 1 calls and you lose The fourth is your EV|player 2 checks After simplifying of the EV formula you should get EV(P2)=(-4x^2+6x-2)/3 EV' = -8x/3 +2 So x=3/4 . This means that player two should bet with 75-100 and player one should call with 84-100 . This is better than my previous attempt .
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#9
09-13-2007, 04:09 AM
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Re: Game Theory Problem Of The Week
Ok here is what I just don't seem to understand.
You apply a formula, get an answer. But do you do anything else to verify it actually answers the question? Your P1 [1,83] fold [84,100] call Your P2 [1,74] Check [75,100] Bet My P1 [1,56] Fold [57,100] Call My P2 [1,11] Bet [12,78] Check [79,100] Bet My P1 vs Your P2 EV = -0.094545 Your P1 vs My P2 EV = -0.118182 My Strategy wins 0.011815 Ante's per hand from yours while rotating positions. Maximal against your P1 "P1MO" [1,65] Bet [66,92] Check [93,100] Bet Maximal against your P2 "P2MO" [1,83] Fold [84,100] Call Your P1 vs Your P2 EV = +0.023636 Your P1 vs P1MO EV = -0.447879 P2MO vs Your P2 EV = +0.023636 Maximal would win 0.2357575 Ante's per hand from your strategies while rotating positions
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