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Ok, here is a toy game featuring a "spite-calling" situation:
Basic game is the same as in Math of Poker, pg 127.
*) Every player is dealt a hand in [0...1].
*) SB ("Pusher") can push or fold
*) If SB pushes, BB ("Caller") can call or fold.
*) If there is a showdown, the player with the higher hand has 2/3 pot equity.
We use stacks of 5BB (SB=0.5, BB=1).
So far, thats just "normal" HeadsUp - and there s no possible spite-calling. After all, we are still in a zero-sum game atm. The NE for this "base game" is: Pusher: 70%, Caller: 56%.
Now lets add the possibility to "spite call":
We now change the game, such that the players will convert their stacks to money after the game, and the players goal is to optimize their $EV. However, the conversion is non-linear. Their stack will be converted to money by payout(chips)=sqrt(chips). (Any strictly growing function will do, as long as it grows "slower than linear". Sqrt is an arbitrary choice.)
This models to some degree the situation of an SNG, because doubling up in chips will now be worth less than double $.
In this modified game, the NE would be:
Pusher: Top 100%, Caller: 8.6%.
We are only 5BB deep, and NE suggests that BB is only calling 8.6% against an ATC push. Alright so far [img]/images/graemlins/smile.gif[/img]
In the plot you can see that the Caller can deal huge "EV-damage" to the pusher, by sacrificing very little EV himself. I think that the NE is unsuitable in this situation, because the caller could clearly "force" the pusher into a more favorable state.
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Great post! If you have the code at hand and it's easy to edit, could you try something:
Iterate through each call range from 0.0 to 1.0 in 0.01 graduations and find the best-response strategy for the pusher along with the EV for the caller vs this best-response strategy. Then plot the EVs for each call range and also find the optimal "spite calling equilibrium" range for the caller.
How does the optimal "spite calling equilibrium" calling/pushing ranges compare to the NE calling/pushing ranges? How do the EV's compare for both players?
Juk [img]/images/graemlins/smile.gif[/img]