Poker And Games

Journal of Behavioral Decision Making
J. Behav. Dec. Making, 23: 335–352 (2010)
Published online 8 July 2009 in Wiley Online Library
(wileyonlinelibrary.com) DOI: 10.1002/bdm.658

Bluffing and Betting Behavior in a Simplified
Poker Game
DARRYL A. SEALE* and STEVEN E. PHELAN
University of Nevada Las Vegas, Nevada, USA

ABSTRACT
A pure-strategy, simplified poker (PSP) game is proposed,where two players draw from
a small and discrete number of hands. Equilibrium strategies of the game are described
and an experiment is conducted where 120 subjects played the PSP against a computer,
which was programmed to play either the equilibrium solution or a fictitious play (FP)
learning algorithm designed to take advantage of poor play. The results show that
players did not adopt thecutoff-type strategies predicted by the equilibrium solution;
rather they made considerable ‘‘errors’’ by: Betting when they should have checked,
checking when they should have bet, and calling when they should have folded. There is
no evidence that aggregate performance improved over time in either condition
although considerable individual differences were observed among subjects. Behaviorallearning theory (BLT) cannot easily explain these individual differences and
cognitive learning theory (CLT) is introduced to explain the apparent anomalies.
Copyright # 2009 John Wiley & Sons, Ltd.
key words experiment; fictitious play; poker; equilibrium solution; learning
algorithm

BLUFFING IN A SIMPLIFIED POKER GAME
Poker has fascinated game theorists from the very birth of the field (Borel,1938; von Neumann &
Morgenstern, 1947). Von Neumann and Morgenstern, in particular, were able to derive an analytical solution
for a simplified poker game that required players to draw a continuous value from a deck in the range [0, 1]
and then bet on whether they had the highest value. Interestingly, the solution counter-intuitively required
players to bet on the highest and lowest values butmake no bet (i.e., check) on intermediate values. Studies on
the same game with human subjects find that they seldom (if ever) play the optimal strategy; appearing
instead to engage in strategy that can best be characterized as some sort of payoff matching (Bearden, SchulzMahlendorf, & Huettel, 2005; Rapoport, Erev, Abraham, & Olson, 1997).
Game theory has a general presumption that even whenhumans do not play optimal strategies, they can
learn them (e.g., Roth & Erev, 1995). Previous studies on probability matching have demonstrated that the
* Correspondence to: Darryl A. Seale, College of Business, University of Nevada Las Vegas, 4505 Maryland Parkway, Las Vegas, NV
89154-6009, USA. E-mail: dseale@unlv.nevada.edu

Copyright # 2009 John Wiley & Sons, Ltd.

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Journal ofBehavioral Decision Making

behavior can be extinguished in favor of the optimal strategy given enough motivation, feedback on
outcomes, and learning time (Shanks, Tunney, & McCarthy, 2002; Vulkan, 2000). However, prior research on
gambling behavior (specifically blackjack) has found that regular casino players do not learn to play the
optimal strategy over time, despite playing several hours aday and investing relatively large sums of money
(Wagenaar, 1988).
The question of whether players can learn to play an optimal strategy in a simplified poker game (and
under what conditions) is thus an empirical question. In this paper, human subjects play a computer opponent
playing either the optimal solution or a more aggressive strategy that attempts to exploit sub-optimal play.
Our gamesimplifies the decision tasks observed in previous experimental studies of bluffing in poker games
by requiring only pure strategies in equilibrium rather than mixed strategies. Games with pure strategy
equilbria give decision makers their best chance to learn optimal play and aid experimenters in characterizing
deviations from such play. In line with previous research, we expect that players...