1. On Von Neumann Pokerwith Community Cards Reto Spöhel Joint work with Nicla Bernasconi and Julian Lorenz TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAA

3. Outline Introduction von Neumann Poker von Neumann Poker withcommunity cards Outlook: the Newman model Conclusion

4. Research on poker Thegame of poker has beenstudiedfrommany different perspectives: game-theory[this talk] artificalintelligence (heuristics) machinelearning (opponentmodeling) behaviouralpsychology etc. In thegame-theoreticapproach, oneassumes best play for all playersinvolved. allowsdevelopment and application of mathematicaltheories neglectsmanyotherfactors

5. Game-theoreticresearch on poker almostexclusivelyfortwo-playergame twomainlines of attack: simplifiedmodels of „real“ poker e.g., 2 suitswith 5 cards each solutionbybrute-forcecalculation; lots of computational power needed moreabstractmodels, whichhopefullycapture essential features of poker[this talk] e.g., handsarenumbersu.a.r. from [0,1] hopefullyanalyticallysolvable mostimportantmodel: von Neumann poker

6. Von Neumann Poker Pchipsare in thepot at thebeginning. X and Y are dealt independent handsx,y2[0,1] u.a.r. X maymake a bet of aor pass („check“). If X checks, bothhandsarerevealed („showdown“), and theplayerwiththehigher hand winsthepot. If X makes a bet, Y caneithermatchthe bet („call“) orconcedethepot to X („fold“). If Y folds, X winsthepot (and gets his bet back). If Y callsX‘s bet, bothhandsarerevealed and theplayerwiththehigher hand winsthepot and thetwobets. In thefollowingweassumeforsimplicityP = a = 1.

7. Von Neumann Poker Itseemsthat X has an advantage, since Y canonlyreact. So howshould X play to maximize his expectedpayoff? Clearly, alwayscheckingguaranteeshim an expectedpayoff of P/2 = 1/2. Similarly, X cannothopefor an expectedpayoff of morethan P=1, since Y canalwaysfold.

8. Von Neumann Poker At firstsight, onemightguessthat X should bet thebetter half of his hands, i.e., iff x ¸ 1/2. However, once Y realizesthatthisisX‘sstrategy, he will onlycallwithhands y ¸ 2/3, sincethen he wins P+a=2 chipswithprobability at least 1/3 he loses a=1 chipswithprobability at most 2/3 i.e., „thepotoddsare in his favor“. 1 call bet 2/3 1/2 fold check 0 x y

9. Von Neumann Poker X‘sexpectedpayoffcanbefoundbyintegratingoverthehands of x and y and is: P ¢ 1/8 + P ¢ 1/2 ¢ 2/3 + (P+a) ¢ 1/18 – a ¢ 1/9 = 1/8+1/3 = 11/24 < 1/2  X loses money! y 1 -a call bet-call bet P+a 2/3 2/3 0 1/2 bet-fold check fold check P P 0 x 0 x y 1/2

10. So X has no advantageover Y and shouldnever bet? NO! Withthepreviousstrategy, most of X‘s good handsgo to wastebecause Y just folds. However, X caninduce Y to callmoreoftenbyincludingbluffs in his strategy! von Neumann gave an equilibrium pair of strategies Y‘sstrategyis best response to X‘sstrategy X‘sstrategyis best response to Y‘sstrategy von Neumann, 1928 X can achieve an expected payoff of 5/9, which is optimal. Von Neumann Poker

12. x0 = 1/9

13. y1 = 5/9

14. x1 = 7/9check x0 fold bluff-bet 0 x y

16. x0 = 1/9

17. y1 = 5/9

18. x1 = 7/9check x0 fold bluff-bet 0 x y

22. x0 = x0(q)

23. y1 = y1(q)

24. x1 =x1(q)x1 y1 check x0 fold bluff-bet 0 x y

28. Y isinducedbythe q to call!

29. For q ¸ q0, there‘s no point in bluffing, since Y will alwayscallanyway.

30. X value-betsmoreoften to protect his good handsfrombeingflippedinto bad hands.x1 y1 x0 q0 = 1/3

32. i.e., when Y has just enoughincentive to callevery bet of X without X wastingmoney on bluffs.5/9 = 0.555… value of thegame 1/2 = 0.5 q0 = 1/3

33. An improvedmodel Wheredoesthediscontinuity at q0 come from? Wefixedthe bet sizea (arbitrarily) beforethegamestarted. Whatifweallow X to look at his hand and then bet anyamounta ¸ 0 he likes? …and withflipprobability q, 0 · q < 1/2 Newman, 1959 Bernasconi, Lorenz, S., 2007+ X can achieve an expected payoff of (16-q)/(28-8q), which is optimal. X can achieve an expected payoff of 4/7, which is optimal.

35. X checksfor 1/7 · x · 4/7

36. Thegreen line isY‘scallingthreshold

39. knowingthis, X neverbets an amountbetween 0 and a0

40. neither as a value bet (iftheoddsare in his favour [x ¸ 1/2], he bets at least a0),

41. nor as a bluff bet (there‘s no value bet forwhich to inducemorecalls).(q = 1/3) bet a a0 = 1/2 hand x, resp. y

45. Questions?