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Article Excerpt
Introduction
A substantial body of evidence indicates that decisions are shaped by a variety of cognitive biases. Hirshleifer (2001) gives an extensive literature survey of psychological influences on investor decisions. Thaler and Johnson (1990) discuss some problems with experimental data, including people's fears that they might lose money because they don't understand the instructions. We use data from an online poker site to investigate whether experienced poker players change their style of play after winning or losing a big pot. Poker is a very attractive source of data because it avoids many criticisms of artificial experiments, for example, that the subjects are inexperienced or that they are inattentive because the payoffs are small. We find that poker players tend to be less cautious after large losses, evidently attempting to recoup their losses quickly.
Behavioral Theories
Poker is a game of uncertainty that involves many unknown factors. Players do not know the cards that will be dealt, what cards their opponents hold, how their opponents will bet, or what their opponents' bets mean. However, poker players can assign probabilities, and the accuracy of these probabilities plays a crucial role in the implementation of a winning poker strategy.
Optimal decisions should focus on the marginal prospective benefits and costs, without regard for past gains and losses. In poker, every hand is new and unrelated to previous hands. In practice, history does matter because (a) players may revise their assessments of their skills and strategy, and (b) outcomes do have psychological effects. We will summarize several different theories and their implied predictions about behavior after big wins and big losses.
Revised Assessment
Updated assessments are consistent with a Bayesian perspective in which players use wins and losses to assess ability. When large sums of money change hands, a player's decisions are validated or challenged and players may feel more confident or less sure of their ability to assign probabilities.
For experienced players, a small number of hands should have little effect on their assessment of their skills. Similarly, List (2004, p. 615) argues that "consumers with intense market experience behave largely in accordance with neoclassical predictions." If so, experienced poker players should be little influenced by wins and losses. But, of course, humans, are not dispassionate Bayesian statisticians, and they often draw conclusions from limited data that should be unpersuasive (Kahneman and Tversky 1972).
If large losses cause poker players to lose confidence, they may subsequently be less certain of their ability to gauge probabilities and play more cautiously. Similarly, after large wins players may be more confident of their probability assessments and play less cautiously. If so, we might expect big losses to be followed by less aggressive play and big wins to be followed by more aggressive play.
On the other hand, calibration studies, which ask individuals to predict the outcome of an uncertain event (such as an election or the performance of a particular stock) and also to estimate the probability that their prediction will be correct, find that people are typically overconfident (Slovic et al. 1976, Odean 1998) and maintain this misperception despite evidence to the contrary by attributing their successes to skill and their setbacks to bad luck (Langer and Roth 1975, Miller and Ross 1975, Fischhoff 1982). If so, wins and losses may matter little.
Prospect Theory and the Break-Even Hypothesis
Many behavioral models are grounded in prospect theory (Kahneman and Tversky 1979, Tversky and Kahneman 1992), which uses a value function v to characterize potential gains and losses (in contrast to a utility function u that values potential levels of wealth) and a decision weight function [pi] that is used in place of probabilities. Thus, if the reference level of wealth is [W.sub.0] and a gamble will result in wealth [W.sub.1] with probability p and wealth [W.sub.2] with probability 1 - p, the value V of this prospect is
V = [upsilon][[W.sub.1] - [W.sub.0]][pi][p] + [upsilon][[W.sub.2] - [W.sub.0]][pi][1 - p]
in contrast to the expected value of utility
E[U] = u[W.sub.1]p + u[W.sub.2](1 - p).
Prospect theory is a general, flexible framework that can accommodate a wide variety of models. In practice, the experiments done by Kahneman and Tversky (1979), Tversky and Kahneman (1974, 1981), and others (typically involving survey questions or small laboratory gambles) support the following characteristics:
1. The value function is S-shaped (convex for losses and concave for gains) because people tend to be risk averse for moderate probability gains (preferring a $50 gain to a 0.50 probability of gaining $100) and risk seeking for moderate probability losses (preferring a 0.50 probability of losing $100 to a $50 loss).
2. The value function is kinked at the origin and gives more importance to a loss than to a gain of the same magnitude, because most people are not attracted to wagers that give them an equal chance of winning or losing $100.
3. The decision weight function treats extremely unlikely events as impossible and extremely likely events as certain. For less extreme probabilities, the decision weight function overweights small probabilities and underweights medium and large probabilities.
These characteristics of the value function and the decision weight function can explain otherwise puzzling behavior such as simultaneously investing conservatively and purchasing insurance and lottery tickets.
The framing of decisions is also thought to be very important (Kahneman and Tversky 1979, Tversky and Kahneman 1981, Barberis and Thaler 2003, Barberis and Xiong 2009). For example, is the reference point equal to the current level of wealth, the level of wealth at the beginning of the day, week, or year, or a target level of wealth?
Figure 1 shows how the reference point affects the value of a prospect. In this example, there is a 0.50 probability of winning $1,000 and a 0.50 probability of losing $1,000, and we assume that the decision weights are equal to the probabilities. If the reference point is [W.sub.0] = 0, the value of the prospect is negative because of the value function's loss aversion. The value of prospect is less than the value of the doing nothing. If, however, the reference point is [W.sub.0] = -$1,000, perhaps because of an earlier loss, then the value of the prospect is higher than the value of -$1,000 because the value function is convex for losses. The value of a 50% chance of -$2,000 and a 50% chance of breaking even is larger than the value of -$1,000.
[FIGURE 1 OMITTED]
There is considerable evidence that behavior is, in practice, affected by sunk costs (Arkes and Blumer 1985)....
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