Behavioural aspects

During recent years, the attention for the integration of psychology into eco-nomics has greatly increased. This has lead to a large stream of literature on behavioural (or psychological) economics. The goal of researchers in this field is to investigate departures from the standard assumptions about human behaviour that are made by economists. To have a concrete frame of reference, we first formulate the classical model of individual choice under uncertainty.

Economic agents are assumed to maximize the expected value of a utility

func-tion of the form maxx∈X

X

s∈S

π(s)U (x|s), (2.2)

where X is the agent’s set of possible choices, S is the state space, π(s) are the agent’s subjective beliefs updated using Bayes’ rule, and U is a utility function that represents the agent’s preferences over all available choices. In the remainder of this section, we discuss some of the psychological phenomena that give rise to alternative models of individual decision making. We use the same division into three categories as Rabin (2002) did in his Alfred Mar-shall Lecture: assumptions about preferences (section 2.4.1.1), heuristics and biases in judgment (section 2.4.1.2) and lack of “stable utility maximization”

(section 2.4.1.3).

2.4.1.1 Assumptions about preferences

The first category of departures from the standard theory consists of attempts to make U (x|s) more realistic. Important lessons in this direction can be learnt from prospect theory, the theory that was introduced by Kahneman and Tver-sky (1979). Prospect theory uses two functions to characterize choices: the value function, which replaces the utility function in standard expected utility theory, and the decision weight function, which transforms probabilities into decision weights. One of the key properties of the decision weight function seems to be important for the behaviour of beginners in games: small prob-abilities are overweighed. As an example, consider video poker players who draw new cards too often, hoping for the royal flush that they will proba-bly receive only once in their lifetimes. They overestimate the probability of receiving such a good hand.

The value function in prospect theory has three important characteristics:

1. changes in wealth are important, not final asset positions;

2. the function is S-shaped; it is concave for gains and convex for losses;

3. the part regarding losses is steeper than the gains part: this reflects loss aversion.

The first point is taken into account in the standard game theoretic analysis of casino games, since game rules are nearly always presented in terms of bets and gains. As a consequence, this definition forms the logical basis for the analysis of the beginners in the game. The other two characteristics may give rise to some discussion about a beginner’s strategy, because they may give clues about which strategies are avoided and which strategies will be more attractive in specific games. Epstein (1977) also discusses the characterics of utility functions in the context of gambling.

It is also interesting to note that preferences may change over time. They need not even be constant during an evening in the casino. Participants may consider the history of play relevant for their strategic choices, even in games in which plays are independent of each other. Their decisions may be influenced by losses and gains that were made during previous plays. Thaler and Johnson (1990) present an interesting investigation of the effects of both prior gains and prior losses on preferences. Under some circumstances a prior gain can increase a person’s willingness to accept certain gambles. This phenomenon is called the house money effect. This change in preferences is explained by the tendency of gamblers to perceive a loss as a reduction of previously made gains in this situation. In the case of prior losses, gambles which offer the possibility of breaking even should be treated differently from those who do not. The first case is discussed by Kahneman and Tversky (1979, p. 287).

They conclude that “a person who has not made peace with his losses is likely to accept gambles that would be unacceptable to him otherwise”. Thaler and Johnson (1990) conjecture that it is important in the examples presented by Kahneman and Tversky that the second gamble always offers a possibility to return to the point of departure. If such a possibility is not present, prior losses may often lead to increased risk aversion. The above findings are all phrased in terms of preferences over gambles (prospects), but it is not difficult to apply the results to (preferences over) strategies of a player.

Another class of modifications of the utility function is formed by the alter-native social preferences. The idea underlying these modifications is that self-interest is not the only motivation that individuals use when making choices.

People can also be interested in another person’s well-being. This altruism can be based on the context in which the decision maker is active (see, e.g.,

Bester and G¨uth (1998) for some examples), but it can also be based on con-siderations of fairness or reciprocity, as Rabin (1993) argues. At first sight, things like fairness and altruism seem to be an unlikely explanation for de-viation from rational play in casino games. After all, most participants will have increasing personal welfare as a goal (or at least as a subgoal, besides the utility they may receive from gambling). Most casino games are, possibly apart from some entrance fee, zero-sum. Being altruistic in a zero-sum game is equivalent to being masochistic. A reason why reciprocity may play a role, however, is the following. In more-person casino games, such as poker, it is difficult for a professional player to make a profit sitting at a table with other professionals. When beginners are joining the game, there are possibilities for the professionals to gain by taking advantage of their weaknesses. This way of acting by the advanced players is completely rational: “the best way for one to play a game depends on how others actually play, not on how some theory dictates that rational people should play” (Goeree and Holt (2001, p. 1419)).

If a beginner somehow notices that some of his opponents are playing “against him”, he may see this as a motivation to try to keep them from making profit, instead of focusing on trying to make profit himself. Although it is not im-mediately clear how this effect could be incorporated in a beginner’s strategy, such reciprocity considerations could play a role.

2.4.1.2 Heuristics and biases in judgment

Whereas the first category of departures from the standard expected utility model has to do with taste, the category that we discuss in this section is about mistakes made by the decision maker. These errors include overconfidence and a biased judgment about various game elements, but also the inability to randomize correctly.

The first phenomenon of interest is overconfidence. This reflects the ten-dency of players to overestimate their own abilities, their prospect for success or the probability of positive outcomes. In behavioural economics a large stream of research has been devoted to this subject; see, e.g., Camerer and Lovallo (1999) and Hvide (2002). Overestimating one’s own abilities, relative to the others, is sometimes referred to as the “better than average effect”:

more than half of the people think they will perform better than the average

person. A too positive idea about one’s own skill can also lead to unrealistic optimism about the chances of attaining good outcomes; see, e.g., Weinstein (1980). The combination of these types of errors forms a good explanation of the inexperienced poker players who bet (bluff) too often and with relatively bad hands.

Another thing that forms a problem for inexperienced players, is random-ization. They believe in the “law of small numbers”, as Tversky and Kahneman (1971) phrase it. That is, they wrongfully assume that the pattern of a large population will be replicated in all of its subsets. This is reflected, for ex-ample, in roulette: people expect a black number to come up after a series of red numbers. But it is also applicable in games in which equilibrium play requires mixed strategies. As an example one can think of bluffing elements in poker: with a low hand you often fold, but sometimes you bet to mislead your opponent. Series of decisions that are based on randomization, which should be independent, will often show a negative correlation if the randomization is done by beginners. In this way, beginners become preys for the professionals, because their “random sequences” are predictable. Not only beginners have problems with this aspect of game play, this is a tendency among people in general. Palacios-Huerta (2003) claims that an exception is formed by profes-sional soccer players taking penalties: in his study, he finds that “profesprofes-sionals play minimax”.

A final type of mistakes made by beginners is simply having an incorrect or incomplete image of the game they are playing. They make mental models of the game that need not coincide with the standard game representation, e.g., by a tree or a normal form. People tend to focus on specific strategies for various reasons and often they ignore the payoffs of the opponent.³ The mental model that someone forms of a game will also depend on the way (and order) in which the rules are explained to him. For examples and a more elaborate discussion, we refer to Warglien, Devetag and Legrenzi (1999).

3In general, even if they take the opponent’s payoffs into account, people only do a few steps of iterated reasoning, so that this information is only partly used in strategic decisions;

see, e.g., Camerer (2003, chapter 5).

2.4.1.3 Lack of “stable utility maximization”

The last category of modifications of the standard assumptions is based on psychological findings that suggest that there do not exist well-defined utilities U (x|s) such that behaviour is best described by assuming that people maximize a function of the form that is given in formula (2.2). For an overview of utility theory, including a discussion of the preference relations underlying utility functions, we refer to Luce and Raiffa (1957, chapter 2) or to Fishburn (1970).

An example of a phenomenon that may be relevant for analyzing beginners, is the tendency of people to “rationalize the past” as Eyster (2002) calls it. A past choice that is suboptimal given a current action may not be suboptimal given another current action. If so, then a person can rationalize the past choice by changing his current action; often someone can choose a current action consistent with his past choice having been optimal. In casino games, this phenomenon can be observed when a poker player keeps raising just because he raised the first time, even though his estimates of the winning probabilities might have drastically lowered as a result of the actions of his opponents.

A second issue that keeps inexperienced players from maximizing a formula like (2.2), is the fact that they find it difficult to think through disjunctions:

according to the sure-thing principle (STP), if a person would prefer a to b knowing that X occured, and if he would also prefer a to b knowing that X did not occur, then he definitely prefers a to b. Shafir (1994) reviews a number of experimental studies of decision under uncertainty that exhibit violations of STP in simple disjunctive situations. The author argues that a necessary condition for such violations is people’s failure to see through the underlying disjunctions. In a game theoretic context, this implies that players may not always be capable of looking ahead in game trees. The more complex the game is, the more this will be a problem for a beginner who tries to formulate a good strategy.