[WhiteWolf]

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There is no need to randomize your play. All the betting is done before you receive your cards, there is no chance of making an opponent fold. Your only advantage in this game is in arranging your two card hand more optimally than your opponent.

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Theoretically there can be a rock-paper-scissors component to this game (see below), where your best hand selection depends on what hand your opponent selects, and his best hand selection depends on what you select. You could then enter into a guessing game with your opponent. Multi-level thinking and game-theoretic optimal play via randomization would then come into play.

However, my gut feeling is that this effect would be very small, and only really apply in heads-up situations. Multiway optimal play would almost certainly be very mechanical: All possible hands would have an absolute ranking, and you would just have to memorize the ranking chart and pick the appropriate hand. Your only edge would come from players who did a worse job of memorizing the optimal rankings of the hands. Once again, this is only a gut feeling and not based on any analysis or simulation.

PS: Here's the rock-paper-scissors hold'em situation: You can pick one preflop hand from A[img]/images/graemlins/diamond.gif[/img] K[img]/images/graemlins/heart.gif[/img], 5[img]/images/graemlins/diamond.gif[/img] 5[img]/images/graemlins/heart.gif[/img], or J[img]/images/graemlins/spade.gif[/img] T[img]/images/graemlins/spade.gif[/img]. I'll then pick one of the remaining two and we'll deal out a hold'em hand. No matter what you pick, I'll have the edge on you:

You pick A[img]/images/graemlins/diamond.gif[/img] K[img]/images/graemlins/heart.gif[/img], I'll pick 5[img]/images/graemlins/diamond.gif[/img] 5[img]/images/graemlins/heart.gif[/img]
pokenum -h 5d 5h - ad kh
Holdem Hi: 1712304 enumerated boards
cards win %win lose %lose tie %tie EV
5d 5h 923864 53.95 779367 45.52 9073 0.53 0.542
Ad Kh 779367 45.52 923864 53.95 9073 0.53 0.458

You pick 5[img]/images/graemlins/diamond.gif[/img] 5[img]/images/graemlins/heart.gif[/img], I'll pick J[img]/images/graemlins/spade.gif[/img] T[img]/images/graemlins/spade.gif[/img]
pokenum -h js ts - 5d 5h
Holdem Hi: 1712304 enumerated boards
cards win %win lose %lose tie %tie EV
Js Ts 891897 52.09 807701 47.17 12706 0.74 0.525
5d 5h 807701 47.17 891897 52.09 12706 0.74 0.475

You pick J[img]/images/graemlins/spade.gif[/img] T[img]/images/graemlins/spade.gif[/img], I'll pick A[img]/images/graemlins/diamond.gif[/img] K[img]/images/graemlins/heart.gif[/img]
pokenum -h ad kh - js ts
Holdem Hi: 1712304 enumerated boards
cards win %win lose %lose tie %tie EV
Ad Kh 1003559 58.61 701504 40.97 7241 0.42 0.588
Js Ts 701504 40.97 1003559 58.61 7241 0.42 0.412