[ QUOTE ]
Assume that we're down to the last two players of a SNG. The payout is 0.5 to 1st place and 0.3 to 2d place. Each player uses an optimal strategy. Is the equity function still linear? What does it look like? Should one player make -CEV plays?
[/ QUOTE ]
(The following assumes the proof I gave is OK and we play with a randomized button.)
Yes, it's linear: at the moment the 3rd place player is decided, if one of the remaining two players has 100x% of the chips and both are playing optimally, his equity is 0.3 + 0.2x .
If the players are using optimal strategies they'll never make -CEV plays, even if their opponent were suddenly to deviate. If one of them deviates to the point where he does make a -CEV play worth -100x% chips, his oppo gains 0.2x equity.
[ QUOTE ]
Assume that
one player gives another player odds in a heads up match. Again, each player uses an optimal strategy. Is the equity function still linear? What does it look like? Should one player make -CEV plays?
[/ QUOTE ]
I interpret this as: if I lose the tourney I lose $10, if I win I get $11, but we still start with equal numbers of tourney chips. Then with optimal play from both players, if I have 100x% of the chips at some point, my equity is 21x-10.
In general one should only ever make -CEV plays if you think it will induce the opponent to make even worse -CEV plays (in total) in later hands of the tournament. Of course a play may be -CEV against one type of oppo while +CEV against another.
Optimal plays will always be >=0 CEV against any oppo.
Marv