This may be illustrated by a consideration of the game of blackjack. In blackjack, the probabilities slowly change as cards are drawn without replacement. As a result, probabilities and values of expected utility may have to be calculated in real time during actual play. Unfortunately, to answer comparatively simple questions about the effects of decision strategies (Greenberg, 1985) or shuffling (Gwynn, 1988) upon the outcomes in blackjack requires extremely lengthy and complex computer simulation. Because people play these games with comparatively little computational effort, it is difficult to believe that these players are performing such complex simulations and calculations in real time before making each bet. Clearly, probabilistic theories fall short of providing a complete explanation for gambling behavior in situations where the odds fluctuate or are unclear.

Practicality of application. Probabilistic theories derive their predictions from the ideal, but unrealistic, situation in which play continues forever, with unlimited bankrolls. Most real-life gambles are infrequent and short-term in nature. Probabilistic theories combine outcome values and probabilities in the long run, but not in the short run (Keren & Wagenaar, 1987; Lopes, 1983). Because gamblers' lives and bankrolls are limited, they are not likely to survive until such long-term outcomes are realized. Because of these limitations, the probabilistic models may not be general enough to ascribe odds and explain play.

Evidential models of choice behavior assume that people make their decisions and gambles on the basis of the available facts. In his theoretical paper on prediction, Cohen (1979) compared the probabilistic theories derived from Pascal and from Bacon (evidential models). Cohen pointed out the limitations of traditional probability theory in predicting people's decision-making behavior. He argued that evidential concepts of odds were more appropriate for describing choice behavior. Whereas probability theory calculates odds on the assumption that all relevant facts are specified in the evidence, evidential functions calculate odds on the extent to which all relevant facts are specified in the evidence. In evidential models, the odds only increase with the weight of evidence. For instance, for a North American 30-year-old man, the probability of his survival until the age of 50 is high from the Pascalian view, whereas from an evidential point of view, the odds are low because the weight of the evidence of age is rather low: The man could be a rock climber or a heavy smoker, and so forth. Unless facts such as these are known, any prediction about whether a 30-year-old man will live for another 20 years is difficult. Cohen suggested that people prefer to base their reasoning upon evidential models rather than upon probability theories, because causal reasoning develops earlier in childhood. Evidential models of choice behavior would suggest that wagers are made on the basis of players' confidence in their prediction of the outcome of play. Those predictions could be based upon probability theory as well as any other variables that players consider relevant.

Pitz's work (Pitz, 1970; Peterson & Pitz, 1988) supported the appropriateness of evidential models of choice behavior. Pitz (1970) suggested that decision makers view and base their decision on the evidence for a particular outcome. For example, Peterson and Pitz (1988) asked participants to estimate the number of games won by a National League baseball team during a 163-game season. They found that participants' confidence in their answers increased positively with the amount of salient evidence they possessed. Similarly, players who were allowed extra time to acquire and evaluate more evidence in computer blackjack increased their bet size and, presumably, their confidence in outcomes (Phillips & Amrhein, 1989).

Pitz (1970) and Peterson and Pitz (1988) also emphasized the need to take into account events seen as relevant by the decision maker, such as events in the decision maker's long-term memory (e.g., habits, expectations, etc.), as well as the salient informational events in the task confronting the decision maker. Such events may be irrelevant in the eyes of the researcher, but they can affect the decision maker's estimation of the likelihood of various outcomes, the decision, and the confidence in the correctness of the decision. For example, Wagenaar (1988) found that blackjack players often deviated from the strategies derived from probabilistic models, despite continued losses. This kind of play can be considered irrational, because it is suboptimal from the perspective of the probabilistic models. In explaining such a "fallacy" on the player's part, Wagenaar (1988) suggested that "a preference for winning does not necessarily entail the mathematical normative concept and the narrow optimal strategy that follows from it" (p. 46) and that one must consider the beliefs and knowledge that the player uses within a game.

Models of Risk-Taking Behavior

Probabilistic and cognitive theories of risk-taking behavior make different ascriptions regarding the importance of short-term fluctuations in the odds, the effects of personal control and involvement, and the role of mediating cognitions. In our study, we compared these models of choice by investigating how wagering is affected by short-term fluctuations of odds, control of a skill-relevant factor, and control of a skill-irrelevant factor. We regarded the amount of the wager as an indication of the degree of confidence in the prediction of an outcome.

We chose blackjack because it is a simple game but still can involve complex decisions. Because the cards are dealt without replacement, exact odds are not available and fluctuate during play. In addition, because they are responsible for making tactical decisions during play, players can affect the odds of winning. The tremendous freedom players have in making decisions may also increase their tendency to attribute the outcome of the game to their skill rather than to the odds. We also included a number of variations upon the game of blackjack in our study, and some of these variations may be relevant or irrelevant to eventual outcomes. The main reason for using a computerized version of blackjack was to enable the manipulation of short-term fluctuations of odds by producing winning and losing streaks; but the task also allowed us to examine the cognitive processes underlying computerized gaming. Even though computerized gaming is becoming increasingly available, most previous studies on gambling have concentrated on more traditional forms of gambling, which may entail different reasoning in the player (Eisler, 1992).

Our predictions may be seen in Table 1. According to probabilistic models of risk-taking behavior, players' conceptions of which wagering strategy should give them maximum return is entirely determined by long-term probabilities of [TABULAR DATA FOR TABLE 1 OMITTED] wins and losses in the game and the weight they assign to these probabilities. More important, such probabilities or weights should not be altered by any short-term fluctuations in odds or changes in amount of control. Because such models do not consider players' reasoning or cognitions (e.g., skill, luck, control of cards, etc.), players' wagers should not be affected by any of the experimental manipulations. Finally, players will attribute the outcomes, no matter what they are, to the long-term probabilities of winning or losing the game. In other words, they will most likely attribute the outcomes to chance-related factors, such as luck, rather than to personal factors, such as ability in a given set of experimental conditions.

On the other hand, cognitive models of risk-taking behavior suggest that the player has different cognitions in different experimental conditions. For instance, short-term fluctuations in odds could affect the decision maker's confidence. The cognitive-irrational models predict that players engage in different irrational cognitions, depending upon the outcome. For instance, in winning streaks, players may think that luck is on their side or that they have figured out a strategy to beat the machine. Such cognitions will lead to an increase in wager. On the other hand, the cognitive-evidential models suggest that the outcomes of previous hands serve as sources of evidence about the game and about players themselves. For example, a losing streak might provide evidence to players that their style of play is inappropriate. Such cognitions also lead to adjustments in wager.

In addition, cognitive models of risk-taking behavior would suggest that personal control and involvement is an important determinant of behavior, either because of the illusion of control it produces or because increased control allows players to exercise their personal strategies. For instance, if players consider themselves to be good players, they may see themselves as having a better chance of winning when they have card control than when they are betting on other players (or other player's strategies). Furthermore, because the cognitive models propose that cognitions underlie gambling, they suggest that players will make a variety of attributions about the outcomes, depending upon whether they win, lose, or can exercise personal control.

Our study distinguished between personal control over skill-relevant dimensions of play and over skill-irrelevant dimensions of play. The ability to request extra cards during play was considered to be a skill-relevant dimension of blackjack. The ability to choose a "dealer" or a gaming machine, although potentially important to players (Eisler, 1992), was considered to be a skill-irrelevant dimension.

Cognitive-irrational models would predict that any extra control should increase a player's confidence in his or her bet and thus would also increase the size of the bet. Such extra control should also interact with short-term fluctuations in odds to affect wagers. A player who is able to choose the "right" dealer in a winning streak will try to exploit an advantage over the dealer by increasing the wager. Players are more likely to attribute the outcome to their ability to exploit the dealer. In contrast, a player who has no choice of dealer will bet more cautiously; such a player will believe that there is no possibility of exploiting any differences between dealers and will, consequently, attribute less of the outcome to his or her ability.

According to cognitive - evidential models, players should be affected only by control over skill-relevant factors, but not by control over skill-irrelevant factors. Card control therefore: affects wagers and attributions to luck and ability. Also, card control should interact with short-term fluctuations in odds. When players have card control, they are more likely to attribute a winning outcome to their ability and a losing outcome to their luck. When players do not have card control, the outcome should have much smaller effects on their attributions. Because skill-irrelevant factors are immaterial, effects involving dealer control should be nonsignificant.

Method

Participants

Participants were 12 undergraduate students at the University of Hong Kong. Their ages ranged from 18 to 25, with a mean age of 20.4. All of them knew the game of blackjack but had no experience in playing the game at casinos. Four of them participated in recreational gambling occasionally. Players received monetary rewards for their participation. The size of the reward was determined by the number of points they had won at the end of the game. Rewards ranged from 20 to 79 Hong Kong dollars (HK$; US$2.5-10), with a mean of HK$37.6 (US$4.82).

Apparatus and Task

A computer program simulated the game of blackjack on an IBM-compatible 486 personal computer. To make the task more realistic, we used rules similar to those used in casino blackjack, although there were some simplifications. The player was presented with a green screen, at the center of which were written "Blackjack pays 3/2" and "Dealer must stand on 17 and draw on 16." Minimum and maximum bets (1 and 9) were clearly displayed in the upper right corner. Card totals were calculated as in blackjack; for example, ace counted as 1 or 11, and the picture cards - jack, queen, and king - counted as 10. However, in a simplification of the rules, there was no provision for splitting, doubling, and insurance, and standoffs were not allowed when the player drew blackjack. Cards and totals for the dealer were displayed at the top of the screen; the player's cards and totals were displayed at the bottom. The amount bet, the outcome, the amount won on that hand, and total winnings were displayed at the very bottom of the screen.

The play followed that of casino blackjack. A blank screen signified that a hand was about to commence. The complete screen then followed, and the display "Dealer is Andrew" or "Dealer is Andrew, Push any Button for Bruce" was presented. The complete screen was then displayed, and a "Place your bets" prompt was clearly displayed. Players entered a number from 1 to 9 to indicate a bet. The player was given 1.5 s in which to place a bet. If the time expired without a bet, the computer assigned the minimum bet of one. Two cards were then dealt to the player, and then one was dealt to the dealer. The player and then the dealer subsequently drew cards, sat, or went bust.

Each player was given HK$20 (approximately US$2.5) to start the game. Each point won or lost was worth HK$0.1. To make the monetary gain or loss more substantive, we stated that each bet of 1 point produced a win or loss of 6 points. Players were allowed to continue even if they had lost all 20 dollars. Players who had 200 or more points at the end of the session were rewarded with dollars equal to their point total. Those who had fewer than 200 points were paid HK$20 for their participation. However, all players were led to believe at the beginning of the experiment that the amount of money they could keep depended on their final point total and that, if they did not win, they might leave the experiment empty-handed.

Three factors were manipulated in the game, namely, winning and losing streaks, control of cards, and control of dealers.

Winning-losing streaks. The probability of the player's winning a particular hand in this version of blackjack was approximately 47%. To produce short-term fluctuations of the odds, we set the probability of winning for a set of 20 hands at approximately 65% or 35% by biasing some of the cards dealt to the player and the dealer. The first two cards for the player and the first card for the dealer were not rigged. The probability of the player's drawing blackjack was fair and not manipulated. However, the possible totals of the subsequent cards dealt were fixed by the computer program. If the player had blackjack and the computer had an ace or a 10, then the computer could not draw a 10 or an ace for the second card.

Hands were randomly assigned as winners or losers within a set of 20 hands, in proportions to produce winning or losing streaks. For the cards dealt to the player, if the player's hand was to be a loser, the cards dealt to the player were random. If the player's hand was to be a winner, the cards were dealt so that the total could not exceed 21 (i.e., the player could not bust). For the cards dealt to the dealer, if the player's hand was to be a loser, the cards were dealt to the dealer so that the dealer sat on a total equal to or greater than that of the player (without busting). For this computer program, the proportions of blackjacks and standoffs were random, and this affected the proportions of winning hands. The probability of winning across all conditions was 47%. When players were assigned a winning streak, their average probability of winning a hand was 61.3%, and their average probability of losing was 33%. When players were assigned a losing streak, their average probability of winning was 32.8%, and of losing, 63%.

Control of cards. When the player lacked control of cards, "Bet on other player" was displayed at the center of the screen. The other "player" was a computer algorithm that sat on a total of 15 or greater. When the player controlled the cards, "Player controls cards" was displayed. "Another card?" was displayed when the player could draw another card. To receive another card, the player had to press any key on the keyboard within 1.5 s after the last card was dealt.

Control of dealers. When the player lacked control of the dealer, "Dealer is Andrew" was displayed against a green background at the center of the screen. When the player had control over the dealer, "Dealer is Andrew, Push any key for Bruce" was displayed. To choose the other dealer, the player had to press any key on the keyboard within 1.5 s. If a key was pressed, the background of the screen changed from green to blue, and "Dealer is Bruce" appeared on the screen.

Procedure