The Impact of Trembling on Behavior in the Trust Game
James C. Cox Cary A. Deck
Department of Economics Department of Economics
University of Arizona University of Arkansas
October 2002
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Behavior in Trust Games: The Impact of Uncertainty and Intent on Second Movers, Study notes of Acting
The results of experiments investigating behavior in trust games where the first mover's decision is either implemented or not. The study reveals that second movers exhibit a mix of generous and selfish behavior, with some using uncertainty to justify their actions. Researchers attribute motivations to actions in various games, including the investment game and ultimatum game. The document also explains the experimental setup and the observed behavior of decision-makers.
What you will learn
- What motivations are attributed to actions in investment games and ultimatum games?
- What is the significance of uncertainty in trust games for second movers' behavior?
- What is the observed behavior of decision-makers in trust games with trembling?
- How do first movers' decisions influence second movers' behavior in trust games?
- How does the experimental setup of trust games affect decision-makers' behavior?
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The Impact of Trembling on Behavior in the Trust Game
James C. Cox Cary A. Deck
Department of Economics Department of Economics
University of Arizona University of Arkansas
October 2002
The Impact of Trembling on Behavior in the Trust Game
This paper reports the results of experiments designed to investigate behavior in games whensubjects must make an inference about how they arrived at a decision node. Specifically, we investigate a trust game where a random drawing determines if the first mover’s decision isimplemented or if instead the first mover trembles and the other action is undertaken. The results indicate that many second movers are willing to give the first mover the benefit of the doubt, but some use the uncertainty to justify more selfish behavior.movers who are significantly less likely to trust the other person in this situation than in the This pattern is anticipated by first typical sequential trust game.
Keywords: Exogenous Trembling, Intentions, Trust Games
JEL Classification: C70, C91, D64, D
1. Introduction
In an attempt to understand observed behavior, researchers frequently attribute intent to actions. For example, ultimatum game proposers who offer the least possible amount to the second mover are often considered to be selfish or greedy. Previous research has shown that responses to a greedy proposal in ultimatum games can vary depending on the decision context and payoff structure (see for example Hoffman, McCabe, Shachat and Smith 1994). However, such factors do not universally affect behavior (see Cox and Deck 2002a). The behavior of a second mover is a reaction to the first mover’s choice and possibly to inferences about the first mover’s motivation as well. Another type of game in which motivations have been attributed to actions is the often studied investment game of Berg, Dickhaut, and McCabe (1995). In the investment game and the related trust game, the first mover can forgo a certain payoff in order to allow a second person the opportunity to allocate a larger amount of money. The decision by the first mover to forgo the guaranteed payoff is considered a trusting action and the decision by the second mover to share the additional money with the first mover is frequently considered reciprocal. Some studies have examined motives directly by decomposing a game into a triad of
own payoff for any outcome. With repetition, they observed behavior which closely followed the subgame perfect Nash equilibrium prediction for fully rational self-interested agents. They reported that this behavior differed significantly from that observed when subjects actually had complete monetary payoff information for every outcome. Our paper is a further exploration of the decision-making process when agents lack a fundamental piece of information. Specifically we investigate behavior when there is a non- trivial probability that a player trembles, which is implemented by the use of a randomization device. Thus, in order to attribute a motivation to another agent one must first consider the likelihood that the action was deliberate. Following sections describe the experimental design and results. The trust game decision task we investigate has been studied extensively and we draw upon this existing body of literature to further interpret our results. A final section contains concluding remarks.
2. Experimental Design
A total of 48 subject pairs played the trust game shown in Figure 1. Subjects were recruited from undergraduate classes at the University of Arkansas and were paid a $5 show up fee in addition to the payment associated with the outcome of the one shot game. Each subject participated in only one session and in one role. Upon entering the Walton Research Laboratory, the 12 students participating in a session were seated at computer terminals with privacy dividers and read the computerized directions.^1 After all subjects had finished the directions, each person was asked to complete a comprehension handout, which was subsequently checked by the experimenter for correctness. 2 Any subject making an incorrect response on this handout was given an oral and written description of the decision task by the experimenter. Throughout the experiment, neutral language was used in describing the task: the extensive form game was called a “decision tree”; players were referred to as “decision makers”; and so on.
Figure 1: Extensive Form Trust Game 1
2
$$ 400
$$ 2515
(^11) $$ 1010
22
$$ 400
$$ 2515
$$ 1010
Once all of the handouts had been checked, a sheet of additional directions was distributed to each participant. This sheet read as follows. Additional Directions: Once a decision-maker 1 has made a decision by clicking on a branch and pressing send, that decision-maker 1 will be prompted by the computer to pick a number between 1 and4 including 1 and 4. After all decision-maker 1s have selected a number, the experimenter will randomly draw a ball from a bingo cage. If the number the experimenter drawsmaker 1’s decision will remain unchanged does not match the number decision-maker 1 selected, then decision-. However, if the number drawn by the experimenter is the same as the number selected by decision-maker 1, then decision- maker 1’s choice will be reversed by the computer. Decision-maker 2 will never knowthe number selected by the decision-maker 1 counterpart.
After these additional directions were read aloud, subjects were able to ask questions about this procedure. Also, the bingo cage and the numbered balls were shown to the subjects, as was a trial drawing from the bingo cage. Subjects were then prompted by the computer to enter their names. At this point the subjects were shown the game which they were to play one time. Subjects randomly assigned the role of decision-maker 1 selected an action and were then prompted to pick a number between 1 and 4. Next, the drawing was held in the presence of the subjects and then the decision-maker 2’s observed the action attributed to their counterparts. Hence a decision-maker 2, who found himself selecting between keeping the entire $40 or keeping $
trust and dictator game experiments using the same money payoffs and social-distance payoff protocol.
3. Results
Of the seventeen decision maker 2s who had to make a choice, eight acted selfishly. This 53% rate of acting generously is between the 64% rate observed in sequential play of the trust game and 33% rate observed in the dictator game. There is marginally significant evidence that the ($15, $25) outcome was observed more frequently with trembling than in the dictator game (p-value = 0.105 in the one tailed z-test of equal proportions). However, one cannot reject the null hypothesis that decision-maker 2 behavior does not differ when decision maker 1’s action may or may not have been intentional, in favor of the one sided alternative that selfish behavior is observed less frequently with trembling than in the sequential trust game (p-value from z-test is 0.232). This suggests that many, but not all, second movers that condition their responses on decision-maker 1 behavior are willing to give the other person the benefit of the doubt. Given the 25% chance that the computer will reverse a decision-maker 1s choice, it is optimal for a decision-maker 1 to select his or her preferred action (see Appendix 3). Also, decision-maker 1 behavior in the trembling game should not differ from that observed in the standard sequential game unless they believe that decision-maker 2 behavior will change (see Appendix 3). Even though a smaller percentage of decision-maker 2s acting generously is likely to be observed with trembling, previous experimental evidence suggests that decision-maker 1s have difficulty in anticipating decision-maker 2 behavior. Cox and Deck (2002a) found that half of decision maker 1s opted for the equal split of $20 rather than trusting their counterpart under both a single-blind and a double-blind payoff protocol. However, that treatment did significantly alter decision-maker 2 behavior. Instead of two-thirds of the decision-maker 2s acting generously, a reversal occurred: two-thirds of the subjects acted selfishly when their personal action was unknown even to the experimenter.
We observe that decision-maker 1s did anticipate a behavioral shift towards selfishness on the part of decision maker 2s associated with the introduction of trembling. Of the 47 observed decision-maker 1s, 32 chose the ($10,$10) outcome and only 15 trusted the second mover. 4 This 32% trusting rate is in direct contrast to the fairly robust 50% rate found previously. This difference is significant, with a p-value of 0.028, showing that the null hypothesis of equal rates can be rejected in favor of the one-sided alternative using a z-test. This general pattern of behavior was also observed, at least nominally, by Dufwenberg, Gneezy, Güth, and van Damme (2000). Their experiment included a treatment in which the second mover was told that the multiplier was 3 and another treatment in which the second mover was informed that the multiplier would be 2 or 4, each with probability ½. Second movers in the stochastic-multiplier treatment allocated a lower percentage of the surplus to the first mover even though the expected multiplier was the same in both cases. 5 Correctly anticipating this, first movers in the known-multiplier treatment were less trusting, as measured by the percentage of their endowments sent to the second movers.
4. Conclusion
Often, when people make decisions they have incomplete information and therefore have to make inferences about what they observe. This feature of decision-making has two important consequences for studying economic behavior. First a person has to consider, not how his action will be interpreted, but rather how a noisy signal of his action will be interpreted. Second, a person must decide how to react to an action that may or may not have been intentional. This differs from standard laboratory experiments in which subjects have complete information about the game structure. To begin study of decisions with this realistic component, we make use of the extensive literature on trust games. Authors of previous studies with the same stakes used here, $40 per
References
Berg, Joyce, John W. Dickhaut and Kevin A. McCabe, “Trust, Reciprocity, and
Social History,” Games and Economic Behavior , X(1995),122-42.
Cox, James C., “Trust, Reciprocity, and Other-Regarding Preferences: Groups vs.
Individuals and Males vs. Females,” in Advances in Experimental Business
Research , Rami Zwick and Amnon Rapoport, eds. (Boston, MA: Kluwer
Academic Publishers, 2002a).
Cox, James C., “How to Identify Trust and Reciprocity.” Working Paper, University of
Arizona. 2002b.
Cox, James C. and Cary A. Deck, “On the Nature of Reciprocal Motives.” Working
Paper, University of Arkansas. 2002a.
Cox, James C. and Cary A. Deck, “When are Women More Generous than Men?”
Working Paper, University of Arkansas. 2002b.
Deck, Cary A., “A Test of Behavioral and Game Theoretic Models of Play in Exchange
and Insurance Environments,” American Economic Review , XCI(2001), 1546-55.
Dufwenberg, Martin, and Uri Gneezy, Werner Güth, and Eric van Damme, “Direct vs
Indirect Reciprocity: An experiment,” Homo Oeconomicus XVIII(2001), 19-30.
Falk, Armin, Ernst Fehr, and Urs Fischbacher, “On the Nature of Fair Behavior,”
Economic Inquiry , forthcoming.
Güth, Werner, Steffen Huck and Wieland Müller, “The Relevance of Equal Splits in
Ultimatum Games,” Games and Economic Behavior , XXXVII(2001), 161-9.
Hoffman, Elizabeth, Kevin A. McCabe, Keith Shachat, and Vernon L. Smith,
“Preferences, Property Rights, and Anonymity in Bargaining Games,” Games and
Economic Behavior ,
VII(1994), 346-80.
McCabe, Kevin A., Stephen J. Rassenti, and Vernon L. Smith, “Reciprocity, Trust, and
Payoff Privacy in Extensive Form Bargaining,” Games and Economic Behavior ,
XXIV(1998), 10-24.
McCabe, Kevin A. and Vernon L. Smith, “A Comparison of Naïve and Sophisticated
Subject Behavior with Game Theoretic Predictions,” Proceedings of the National
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Appendix 1. Subject Directions
Appendix 3. Theoretical Analysis of Decision-Maker 1 Choices
Let u(x,y) denote decision-maker 1’s utility from the outcome where decision-maker 1 receives $x and the other person receives $y. We assume that no player is indifferent between his or her two alternative actions. Also, let λ be player 1’s subjective probability that player 2 will choose
the outcome (15,25) if player 2 observes a trusting action in the tremble game and let α be the probability that player 1 does not tremble.
Proof that Player 1 should reveal truthfully Suppose that player 1’s preferences are such that u(10,10) > λu(15,25) + (1- λ)u(0,40). That is, player 1 prefers that the second mover not have an opportunity to make a decision. From selecting the action leading to (10,10), player 1 would receive an expected utility of αu(10,10) + (1- α)(λu(15,25) + (1- λ)u(0,40)). From selecting the trusting action, player 1’s
expected utility would be (1-α)u(10,10) + α(λu(15,25) + (1- λ)u(0,40)). Thus this agent should
select the action leading to (10,10) if αu(10,10) + (1- α)(λu(15,25) + (1- λ)u(0,40)) > (1-
α)u(10,10) + α(λu(15,25) + (1- λ)u(0,40)). Which can be rewritten as (2α-1)u(10,10) > (2α-
1)(λu(15,25) + (1- λ)u(0,40). This can be reduced to u(10,10) > λu(15,25) + (1- λ)u(0,40) as
long as 2α-1>0 or α > ½. Given that α= ¾ in the experiments a player 1 who prefers the (10,10) outcome should select that action. As the preference structure in the preceding argument can be replaced by a strict preference for trusting, all player 1s should truthfully reveal. □
Proof that player 1 behavior differs between the trembling game and the standard sequential game only if player 1’s belief about player 2’s behavior differs between the games From the preceding proof, player 1 should select the action that truthfully reflects her preferences if α > ½, given her beliefs represented by the subjective probability λ. In the
trembling game, α = ¾ and in the standard sequential version of the game, α = 1. Let γ be player 1’s belief that player 2 will choose the outcome (15,25) if player 2 observes a trusting action in the standard sequential game. Consider the case of a person who chooses (10,10) when α = ¾
and trusts when α = 1. For this individual it must be that u(10,10) > λu(15,25) + (1- λ)u(0,40)
from above and that u(10,10) < γu(15,25) + (1- γ)u(0,40). The only way these two inequalities
can hold simultaneously is if λ ≠ γ. By symmetry an individual who trusts when α = 1 and
chooses (10,10) when α = ¾ must also believe the probability that player 2 will select (15,25) differs between the standard trust game and the tremble treatment. □