Question

What should a poker player do? bluff on every hand? never bluff? optimize the mix of bluffing? Or rely on flair, not on mixed strategies? Explain briefly.

Answer 1

Poker is an interesting testsbed for artificial intelligence research. It is a game of imperfect information, where multiple competing agents must deal with probabilistic knowledge, risk assessment, and possible deception, not unlike decisions made in the real world. The so-called “opponent modelling” is another difficult problem in decision-making applications, and it is essential to achieve high performances in poker. Moreover, poker has a rich history of study in several academic fields. Economists and mathematicians have applied a variety of analytical techniques to poker-related problems. For example, the earliest investigations in game theory, by luminaries such as John von Neumann and John Nash, used simplified poker to illustrate the fundamental principles.

There is an important difference between board games and popular card games like bridge and poker. In board games, players have complete knowledge of the entire game state since everything is visible to all participants. In contrast, bridge and poker involve imperfect information since the other players’ cards are not known.

From a computational point of view, it is important to distinguish the lack of information from the possibility of chance moves. The former involves uncertainty about the current state of the world, in particular situations where different players have access to different information. The latter involves only uncertainty about the future, uncertainty which is resolved as soon as the future materializes. Perfect and imperfect information games may involve an element of chance; examples of games from all four categories are shown in Table 1.

Perfect information Imperfect information No chance Chess Inspection game Chance Monopoly Poker

Table 1

The presence of chance elements does not need major changes to the computational techniques used to solve a game. In fact, the cost of solving a perfect information game with chance moves is not substantially greater than solving a game with no chance moves. By contrast, the introduction of imperfect information increases the complexity of the problem.

Due to the complexity (both conceptual and algorithmic) of dealing with imperfect information games, this problem has been largely ignored at the computational level until the introduction of randomized strategies concept.

Once randomized strategies are allowed, the existence of “optimal strategies” in imperfect information games can be proved. In particular, this means that there exists an optimal randomized strategy for poker in the same way as there exists an optimal deterministic strategy for chess. Indeed, Kuhn showed for a simplified poker game that the optimal strategy does use randomization.

The optimal strategy has several advantages: the player cannot do better than this strategy if playing against a good opponent; furthermore, the player does not do worse even if his strategy is revealed to his opponent; that is, the opponent gains no advantage from figuring out the first player’s strategy.

Another interesting result of such researches is the existence of an optimal strategy for the gambler in poker game. As first observed in a simple poker-like game by Kuhn [19], behaviors such as bluffing, that seem to arise from the psychological makeup of human players, are actually game theoretically optimal.

One of the earliest and most thorough investigations of poker appears in the classical treatise on game theory “Games and Economic Behavior” by von Neumann and Morgenstern [20], where a large section was devoted to the formal analysis of “bluffing” in several simplified variants of a two-person poker game with either symmetric or asymmetric information.

Indeed, the general considerations concerning poker and the mathematical discussions of the different versions of the game were carried out by von Neumann as early as 1926. Recognizing that “bluffing” in poker “is unquestionably practiced by all experienced players,” von Neumann and Morgenstern identified two reasons for bluffing. “The first is the desire to give a “false” impression of strength in “real” weakness; the second is the desire to give a “false” impression of weakness in “real” strength”.

Solutions to these simplified poker-like games as well as a large class of both zero-sum and nonzero-sum games were unified by the concept of mixed strategy, a probability distribution over the player’s set of actions. The importance of mixed strategies to the theory of games and its applications to the social and behavioral sciences stems from the fact that for many interactive decision processes there can be no Nash equilibria in pure strategies.

Using randomization and adaptive learning as key concepts to modelize into a computational scaffolding of the cognitive processes, we believe that this area of research is more likely to produce insights about superior cognitive strategies because of their intrinsically structures. Finally, comparing them with the real human strategy, it is possible both to investigate the role of environmental factors on cognitive strategies development and to validate some theoretical psychological assumptions.