Unformatted text preview: PSYCHOLOGICAL SCIENCE Research Report
EFFECTS OF FINANCIAL INCENTIVES ON THE BREAKDOWN
OF MUTUAL TRUST
James E. Parco,1 Amnon Rapoport,1 and William E. Stein2
1 Department of Management and Policy, Eller College of Business and Public Administration, University of Arizona, and
Department of Information and Operations Management, Mays College of Business, Texas A & M University Abstract—Disagreements between psychologists and economists about
the need for and size of ﬁnancial incentives continue to be hotly discussed. We examine the effects of ﬁnancial incentives in a class of interactive decision-making situations, called centipede games, in which
mutual trust is essential for cooperation. Invoking backward induction,
the Nash equilibrium solution for these games is counterintuitive. Our
previous research showed that when the number of players in the centipede game is increased from two to three, the game is iterated in time,
the players are rematched, and the stakes are unusually high, behavior
approaches equilibrium play. Results from the present study show that
reducing the size of the stakes elicits dramatically different patterns of
behavior. We argue that when mutual trust is involved, the magnitude of
ﬁnancial incentives can induce a considerable difference.
Psychologists and economists agree that a wide range of incentives
can affect human decision behavior in the laboratory. However, they
often disagree on the methodological and operational aspects involved
in modeling and studying these incentives (Zwick, Erev, & Budescu,
1999). Economists almost always use ﬁnancial incentives. In contrast,
most psychologists who study judgment and decision making (JDM)
use only hypothetical payments. A recent investigation of a 10-year
sample of empirical studies published in the Journal of Behavioral
Decision Making showed that only 48 of the 186 studies (26%) employed ﬁnancial incentives. As Hertwig and Ortmann (2001) pointed
out, this percentage very likely overestimates the use of ﬁnancial incentives (strictly deﬁned as performance-based monetary payments)
in psychological research.
The discussion of the methodological differences in the way psychologists and economists design and conduct their experiments often
focuses on the presence or absence of ﬁnancial incentives (e.g., Camerer, 1997; Hertwig & Ortmann, 2001; Zwick et al., 1999). Important
as this issue is, there is evidence that in assessing the validity of JDM
experimental research and the potential applications of its ﬁndings, it
is the magnitude of ﬁnancial incentives, rather than their presence, that
mostly matters (Camerer & Hogarth, 1999; Gneezy & Rustichini,
2000). Even when they are contingent on performance, low payments
may not be sufﬁcient to overcome the effects of habits, traditions, social norms, moral values, and various emotions that, although recognized as important, are presently excluded from most of the theories of
individual and interactive decision making. Budget considerations
typically impose a severe constraint on subjects’ payments. In the
United States, most JDM experiments that pay their participants limit
individual payment to between $10 and $25 per 2-hr session. It is
quite possible that for many experiments these ﬁnancial incentives are
sufﬁciently strong to overcome (at least in part) intrinsic, social, and Address correspondence to James E. Parco, 405 McClelland Hall, Department of Management and Policy, Eller College of Business and Public Administration, The University of Arizona, Tucson, AZ 85721; e-mail: [email protected] 292 Copyright © 2002 American Psychological Society other nonmonetary motivations. Nevertheless, for generalizing the
laboratory results to real-life situations, it is important to break away
from the customary budget constraints. The major purpose of the
present study was to test the hypothesis that the magnitude of ﬁnancial
incentives in a class of interactive situations involving mutual trust
The most important solution concept for interactive decision making in noncooperative games is the Nash equilibrium—an n-tuple of
strategies with the property that none of the n players can beneﬁt by
unilateral deviation. Put differently, under equilibrium play, each strategy is a best response to the strategies of the remaining n – 1 players.
Nash (1950, 1951) proved that every noncooperative n-person game
with a ﬁnite strategy space has at least one equilibrium in either pure or
mixed strategies.1 Experimental studies designed to assess the descriptive power of the Nash equilibrium have been conducted in the past 30
years or so with mixed results.
The most well known example of the failure of the Nash equilibrium
to account for interactive decision behavior is the two-person ﬁnitely iterated Prisoner’s Dilemma (PD) game. When the number of rounds of play
is common knowledge, backward induction—which is the procedure
used to derive the equilibrium solution for the game—prompts defection
of each player in each round, resulting in a unique equilibrium of defection of both players in the ﬁrst round (e.g., Luce & Raiffa, 1957). There
are now hundreds of experiments, with and without ﬁnancial incentives,
that have soundly rejected this prediction. A second, less known, but possibly more damaging example of the paradox of backward induction is
the two-person centipede game, which is the focus of the present study.
This game was ﬁrst introduced by Rosenthal (1981) and subsequently
studied theoretically by Aumann (1992, 1995, 1998), Binmore (1996),
Ponti (2000), Reny (1992), Stalnaker (1998), and many other investigators. Aumann (1992) referred to the game as one of the “disturbing counterintuitive examples of rational interactive decision-making” (p. 219).
The process of backward induction in a two-player centipede game
is straightforward. Consider the example in Figure 1. At each decision
node, a player must decide whether to “stop” (move down) or “continue” (move right). If a player chooses to stop, the game terminates
and the players receive the payoffs identiﬁed at the corresponding terminal node. Otherwise, the game progresses to the next decision node.
At the ﬁnal decision node of the game tree, the decision to stop dominates the decision to continue2 (assuming that $1,000,000 is preferred
to $0). At the next-to-last decision node, assuming that Player 2 will
play rationally on his ﬁnal move, Player 1 should stop, because, by the
same logic, the decision to stop dominates a continue decision, and so
forth. With the same reasoning applying to all moves, the decision to 1. A mixed strategy is a probability distribution deﬁned over the set of pure
2. A decision to continue at the ﬁnal decision node also results in the termination of the game.
VOL. 13, NO. 3, MAY 2002 PSYCHOLOGICAL SCIENCE James E. Parco, Amnon Rapoport, and William E. Stein Fig. 1. A two-person, six-move centipede game (Aumann, 1992). stop at the ﬁrst decision node is the unique Nash equilibrium of the
game. Note that the game can be extended to any ﬁnite number of
moves, with the payoffs becoming astronomical, without changing the
solution. Note, too, that if Player 2 ﬁnds himself at the second decision node, he learns that Player 1 did not make her inductive choice to
stop at the ﬁrst decision node. Therefore, he may consider it possible
or even likely that she will not make her inductive choice during her
next opportunity either. (See Reny, 1992, for further discussion of rationality in the centipede game.) Rationality permits the players to
draw conclusions from past play and form estimates about future play
in any way they want. One requires the considerably stronger assumption of common knowledge of rationality to derive the implication that
the players should stop at each decision node (e.g., Aumann, 1995).
The backward-induction argument is not restricted to two players,
nor is the centipede game limited to only two players. In a previous
study (Rapoport, Stein, Parco, & Nicholas, 2000; hereafter RSPN), we
extended the centipede game by adding a third player and experimentally implemented it for very high stakes (see Fig. 2). We compare the
results of the present study to those of RSPN to assess the effects of
the magnitude of the ﬁnancial incentives on behavior. This comparison is straightforward, as the two studies were identical in terms of the
population of participants and all the other details of the experimental
design. The only exception was the size of the payoffs (compare Figs.
2 and 3). Our major ﬁnding is that decreasing the stakes produces dramatically different results. METHOD
Thirty undergraduate students from the University of Arizona volunteered to participate in a decision-making experiment for monetary
payoff contingent on performance. Fifteen participants, both males
and females, took part in each of two separate sessions. Each session
lasted approximately 75 min. Procedure
Two separate experimental sessions using different participants
were conducted on separate days at the Economic Science Laboratory
at the University of Arizona. At the beginning of each session, the 15
participants were seated at separate cubicles apart from one another,
each containing a networked computer and set of written instructions,
VOL. 13, NO. 3, MAY 2002 and proceeded to read the instructions at their own pace. (A complete
set of instructions can be found at http://www.eller.arizona.edu/~map/
research.) Any form of verbal communication was strictly forbidden.
Each session consisted of 60 trials. On each trial, the participants
were randomly divided into ﬁve 3-person groups, and within each
group randomly assigned to the roles of Players 1, 2, and 3. At the
start of each trial, all the participants were individually informed of
the trial number and assignment of roles. If a participant assumed the
Player 1 role, the branches on the game tree displayed on his or her
monitor were enabled, allowing the participant to make a decision. Simultaneously, Players 2 and 3 of the same group viewed an identical
screen without the ability to make a decision. Decisions were made by
clicking on either the “stop” (down) branch or the “continue” (right)
branch emanating from the current decision node on the game tree.
When selected, the branch turned color, and a “commit” button appeared on the screen, requiring the participant to conﬁrm his or her decision. Once the decision was conﬁrmed, the game trees on the
monitors of the other two group members were updated, identifying
the selected branch by changing its color. If the decision resulted in
continuation of the game, the next player in the sequence (1, 2, 3, 1, 2,
. . . ) was enabled to make his or her choice in exactly the same manner, while the other two (inactive) players only viewed the updated
game tree on their monitor. If the decision resulted in termination of
the game, all three group members were immediately informed that
the trial was completed. Participants were informed of only the outcomes of the games that they actually played.
The individual payoff was not cumulative. Rather, it was based on the
outcomes of three trials that were chosen randomly by the computer prior
to the start of the experiment. (Participants were informed of the numbers
of these trials only after completing all 60 trials.) Individual payments in
Session 1 varied from $0.72 to $27.20, with a median of $3.54. Individual
payments in Session 2 varied from $0.13 to $12.97, with a median of
$1.52. Because of the low payoffs, each subject received an additional
subsidy of $5.00 (in addition to the $5.00 show-up fee).
The experimental design of the present study (see Fig. 3) was identical to that of the RSPN study (see Fig. 2), differing only in the number
of sessions (four in the RSPN study vs. two in the present study) and the
size of the stakes.3 The stakes in the present study were smaller than 3. To reduce costs, we informed participants in the RSPN study that they
would receive only 50% of their individual earnings in three trials randomly
selected before the experiment started. 293 PSYCHOLOGICAL SCIENCE Financial Incentives Fig. 2. The three-person, nine-move centipede game used in Rapoport, Stein, Parco, and Nicholas (2000). those in the RSPN study by a factor of 100. If all three payment trials in
the present experiment were to terminate at the ﬁnal decision node (with
Player 3 making a stop decision), the maximum payoff per session
would have been $460.80, averaging $30.72 per participant. The corresponding maximum payoff per session in the RSPN experiment would
have been $23,040.00, averaging $1,536.00 per participant. RESULTS
Table 1 (top panel) presents the proportions of games ending at
each of the nine terminal nodes. Because of the random assignment of
the 15 players to ﬁve 3-player groups and the possibility of rematching
with 1 or more players, trials are not independent. In this “playing
against the ﬁeld” design, it is the population rather than the individual
participant or 3-player group that is the unit of analysis. Therefore,
Table 1 presents the results for each session separately, as well as the
combined results across the two sessions. Within each session, the results are combined across the ﬁve 3-player groups in each trial and the
60 trials, for a total of 300 games. For comparison purposes, the results of the RSPN study are displayed (for each session separately and
across sessions) in the lower panel of Table 1.
Several conclusions can be drawn from Table 1:
• All the nine decision nodes in each of the two sessions of the present
study were reached at least twice (2/300 .007).
• In both sessions in the present study, the proportion of stop decisions ﬁrst increased, reaching a peak at the middle of the second
round (ﬁfth decision node), and then decreased. • The patterns of behavior in the present and previous RSPN studies
are entirely different. Whereas only 2.5% of all the games in the
present study ended immediately with a stop decision at the ﬁrst decision node, the corresponding result for the RSPN study (39.2%) is
15 times higher. Whereas only 25.2% of the games in the present
study ended in Round 1 (terminal nodes 1, 2, and 3), with one of the
three group members stopping at the ﬁrst opportunity, the corresponding result for the RSPN study is 83.4%. And whereas the proportion of stopping in the present study ﬁrst increased, from 2.5% to
26.3%, and then decreased, in the RSPN study the proportion of
stopping reached its maximum of 39.2% at the ﬁrst decision node
and then decreased steadily.
Table 2 presents the (inferred) conditional probabilities of a stop
decision at decision node j, denoted by pj. The table supports the conclusions already drawn from Table 1, namely, that the patterns of behavior in the present and RSPN studies are very different. In the
present study, the values of pj increase in j. In contrast, the conditional
probabilities in the RSPN study are remarkably stable across decision
nodes j 1 to j 8. Table 2 shows that this result holds not only for
the summary results, but also for each session separately.
The major reason for the difference between the two studies is the
dynamics of play: convergence to equilibrium in the RSPN study in
contrast to no evidence for learning in the present study. Figure 4 exhibits the arithmetic moving average (in steps of 5) of the conditional
probabilities pj (j 1, 2, 3) for the two sessions of the present study.
Within each session, the probabilities are presented separately for the
three decision nodes in Round 1, namely, 1, 2, and 3. The ﬁgure shows Fig. 3. The three-person, nine-move centipede game used in the present study. 294 VOL. 13, NO. 3, MAY 2002 PSYCHOLOGICAL SCIENCE James E. Parco, Amnon Rapoport, and William E. Stein Table 1. Proportion of games ending at each terminal node
Across sessions N 1 a 300
600 2 3 4 5 6 Low-pay 9-move game (present study)
.027 .043 .093 .240 .263 .227
.023 .067 .250 .243 .263 .097
.025 .055 .172 .242 .263 .162 300
1,200 High-pay 9-move game (RSPN)c
.463 .317 .110 .050 .027
.393 .277 .157 .087 .030
.303 .280 .187 .093 .053
.407 .257 .183 .077 .037
.392 .283 .159 .077 .037 .020
.023 7 8 9 .073
.010 a Number of games (ﬁve groups of 3 randomly matched players per trial participating in 60 trials).
A single Player 3 continued at the 9th decision node.
RSPN Rapoport, Stein, Parco, and Nicholas (2000).
b very little evidence for learning across the 60 trials. Across both sessions and a relatively large number of trials, there is very little movement, if any at all, in the direction of equilibrium play.
Figure 5 portrays the corresponding results for the high-pay centipede experiment of RSPN. It, too, depicts the arithmetic moving averages separately for each of the sessions of the study. In contrast to
Figure 4, Figure 5 shows a tendency for the three conditional probabilities p1, p2, and p3 to increase sharply across trials. The patterns vary
somewhat between sessions, indicating different population dynamics.
The probability of stopping on the ﬁrst decision node, p1, increases
steadily in Sessions 1 and 4. In Session 2, it reaches a plateau after 20
trials or so and then starts increasing again after Trial 40. In Session 3,
the values of p1 are rather ﬂat for the ﬁrst 45 trials or so, and only then start increasing. The trends in the values of p2 and p3, although in general increasing over time, are not as easily discernible. Except in Session 3, the mean values of p1, p2, and p3 on the last 10 trials are all
above .75. Indeed, during the last 10 trials of Sessions 1, 2, and 4, the
game never progressed beyond the ﬁrst round of play. CONCLUSIONS
The direct comparison of the present low-pay centipede game with
the high-pay centipede game of RSPN exhibits strong evidence, perhaps the strongest documented in the literature, that the magnitude of
ﬁnancial incentives makes a signiﬁcant and substantial difference. Not
only do ﬁnancial incentives matter, but when they are sufﬁciently high Table 2. Inferred conditional probability of stopping on decision node j
Across sessions 7 8 9 Low-pay 9-move game (present study)
300 .03 .04 .10 .29 .44 .69
300 .02 .07 .28 .37 .63 .63
600 .02 .06 .19 .33 .54 .66 .71
1.00b High-pay 9-move game (RSPN)c
.46 .59 .50 .46 .44
.39 .46 .47 .50 .35
.30 .40 .45 .41 .39
.41 .43 .55 .50 .48
.39 .47 .49 .46 .42 .75b
1.00 N a 300
1,200 1 2 3 4 5 6 .60
.44 a Number of games (ﬁve groups of 3 randomly matched players per trial participating in 60 trials).
Based on fewer than 10 observations.
RSPN Rapoport, Stein, Parco, and Nicholas (2000).
b VOL. 13, NO. 3, MAY 2002 295 PSYCHOLOGICAL SCIENCE Financial Incentives Fig. 4. Conditional probability of stopping at decision node j in the low-pay centipede game
(present study). they support Hertwig and Ortmann’s (2001) conclusion that when
learning is possible, monetary payments may bring the decisions
closer to the predictions of the normative models. The generality of
this ﬁnding beyond interactions that involve the dissolution of mutual
trust is a topic for further research.
Acknowledgments—We would like to acknowledge support of this research by a grant from the Hong Kong Research Grants Council to the
Hong Kong University of Science and Technology (Project No. CA98/
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