Game Theory Quiz

[Jazza]

This has been posted in the poker theory forum, but it doesn't look like anyone's gonna try it, so i'll post it here for you guys if you wanna give it a try

The game is pot limit 1-card, the deck consists of 5 cards, the A,2,3,4 and 5.

The ace is low, so the 5 is the nuts.

You each ante $100. There are two betting rounds. It is pot limit, so to start off you can be anywhere from $1 to $200 since you both have $100 each in the pot to start.

Now suppose you are playing some one heads up, you are last to act in both betting rounds, and before the first round your opponent decides to have a little fun and flip his card face up, it's the 4.

What's your strategy, and what minimum EV does your strategy guarantee?

[pzhon]

[ QUOTE ]

You each ante $100. There are two betting rounds. It is pot limit, so to start off you can be anywhere from $1 to $200 since you both have $100 each in the pot to start.

Now suppose you are playing some one heads up, you are last to act in both betting rounds, and before the first round your opponent decides to have a little fun and flip his card face up, it's the 4.

What's your strategy, and what minimum EV does your strategy guarantee?

[/ QUOTE ]
If the player with the 4 is restricted to check/call or check/fold, and you have to bet the full pot, then on the river you offer 2:1 odds so you should bluff 1/3 of the time, and give up the rest of the time. When you bet the river, the player with the 4 has 0 equity. So, similarly, you should bluff once half as often as you bet twice. This suggests the strategy of bluffing 5/12 of the time with 1, 2, or 3, while always value betting the pot with the nuts. If your bluff is called, fire a second barrel 2/5 of the time you are bluffing.

Overall, you bet on the first round 9/16 of the time, and you bet on the second round 6/16 of the time. The player with the 4 is indifferent to calling or folding on the first round, so you expect to win 9/16 of the $200 pot.

It doesn't look like you would gain from betting less than the full pot because you would be able to bluff less frequently. It looks suicidal for the player with the 4 to do anything other than to check/call or check/fold. It's funny that even if you were able to bet 100 times the pot, the player with the 4 would retain some equity, since there are only two betting rounds, and you only have the nuts 1/4=1/2^2 of the time.

[Jazza]

well done

if your opponent bets out on the flop, you can simply use the same strategy and bet the new pot 9/16 of the time, and then bet the second round 3/8 of the time while always betting the 5 as usual, and now your EV will be even greater than the minimal $12.5

bonus points if you can figure out the optimal strategy given that your opponent bets out amount X on the first rounds

anyway i thought it was very interesting that although you have the worst hand 75% of the time, the information you have on your opponent turns this into an +EV situation, i think this is a good example of the well known truth that you cannot be too predictable in big bet poker