The Shortstack Game (Part 3)

Just in case anybody is still reading this...having set up our game and solved for player 2's best response function, we now need to figure out player 1's best reponse function. I ended the last post with a couple of questions to get you started considering player 1's best response function:



What's player 1's best response if player 2 folds every time?



Pretty obvious that if player 2 always folds, player 1 should always raise.



What's player 1's best response if player 2 shoves every time?



It's pretty obvious that if player 2 shoves every time, player 1 should never raise/fold. So depending on his hand, he should either raise/call or fold. If he folds, he gets zero, whereas if he raise/calls, he gets 21.5e - 20*(1-e). Set this expression equal to zero and solve for e, you get 20/41.5. So he should raise/call anytime his hand has more than (20/41.5) equity against a random hand. We already determined in the last post that these hands are 22+,A2+,K2+,Q3o+,Q2s+,J7o+,J3s+,T8o+,T6s+, and 97s+, for 54.8% of all hands. Player 1's average payoff is then:



.452*0 + .548*(.572*21.5 - .428*20) = 2.05



Whereas player 2 has an average payoff of -0.55. Not coincidentally, this is the exact inverse of the payoffs when player 1 raise/calls every time and player 2 best responds.



What if player 2 shoves 22+,A9o+,A5s+,KQo,KJs+ (14.6% of all hands) and folds everything else?



We will have to solve this by "backward induction", considering the last decision first and then using that to figure out the payoffs involved in making earlier decisions. So let's assume player 1 has raised and player 2 has shoved. Which hands should player 1 be calling with? He would be calling 17 to win 24.5, so he needs 24.5*e - 17*(1-e) > 0, or e greater than 40.9%. These hands are: 33+, ATo+, A9s+,KQs (9.7% of all hands). All other hands, had player 1 raised them, would be folded. Player 1's calling range then has 53.8% equity against player 2's shoving range. Now we can consider player 1's decision to raise or fold as his first move. Say he has the most marginal hand with exactly 40.9% equity. Should he raise? His expected payoff to raising is:



.854*1.5 + .146*(.409*21.5 - .591*20) = .84



This is clearly better than 0, the payoff from folding. Also note something else. If player 1 has a hand that he will NOT call a raise with, his payoff to raising is:



.854*1.5 + .146*(-3) = .84



This is also better than folding. That the payoff to raise/calling and raise/folding with the 40.9% equity hand is no coincidence. The 40.9% mark is exactly the point where calling player 2's shove and folding to it are equal in expected value.



Player 1's expected payoff to this strategy is now:



.854*1.5 + .146*(.097*(21.5*.538 - 20*.462) - .903*(1.5)) = 1.1



What is player 1's best response to any given strategy for player 2?



Let f be the % of the time that player 2 folds when player 1 raises. Then player 1's expected payoff of raise/calling is:



1.5*f +(1-f)(-3) = 4.5f - 3



This obviously equals 0 when f equals 2/3. So if player 2 is folding two-thirds of the time or greater, player 1 should raise every time. If player 2 shoves, then player 1 should call if he has greater than 40.9% equity vs. player 2's shoving range.



If player 2 folds less than 2/3 of the time (and therefore shoves more than 1/3 of the time), player 1 should never raise/fold, and should raise/call if:



1.5*f + (1-f)*(21.5e - 20*(1-e)) > 0



Where e is the equity of player 1's hand against player 2's shoving range.



-BRUECHIPS