Public Choice 106: 137–155, 2001.
© 2001 Kluwer Academic Publishers. Printed in the Netherlands.
137
Cooperation in PD games: Fear, greed, and history of play ∗
T.K. AHN1 , ELINOR OSTROM1, DAVID SCHMIDT2, ROBERT SHUPP2
& JAMES WALKER2∗∗
1 Department of Political Science, Woodburn Hall, Indiana University, Bloomington, IN
47405, U.S.A.; (Workshop in Political Theory and Policy Analysis); 2 Department of
Economics, Wylie Hall 105, Indiana University, Bloomington, IN 47405, U.S.A. (Workshop in
Political Theory and Policy Analysis)
Accepted 12 July 1999
Abstract. The impact of the cardinal relationships among pecuniary payoffs, and of social
history and reputation, on the choice of strategies in four one-shot Prisoner’s Dilemma games
is experimentally examined. The results suggest that normalized payoff values linked to “fear”
and “greed” are important as predictors of behavior in the PD games. Success in coordinating
on the payoff dominant equilibrium in previous plays of coordination games also increases
the probability of cooperative play in the PD games. The effect of past play is strongest when
individuals are matched repeatedly with the same person in previous play, as contrasted to
being matched randomly with another player.
1. Introduction
Individuals, in deciding whether to contribute to the provision of a public
good, whether to shirk or work hard on a team project, or whether to dump
industrial chemicals in a fresh water stream or to clean up the effluent, face
tough choices that can often be represented as a Prisoner’s Dilemma (PD)
game. In a PD game, players have a dominant strategy not to cooperate
(called Defect) with others. When all players follow this dominant strategy
they obtain an outcome that is Pareto inferior to the outcome resulting when
players cooperate with one another (Cooperate). Non-cooperative game theory thus leads to a prediction that individuals will contribute suboptimally
to the provision of public goods, will shirk on their fellow team members,
and will dump pollutants in the environment. One commonly stated justifica∗ Presented at the fall meetings of the Economics Science Association, Tucson, Arizona,
October 1998. The authors would like to thank the American Academy of Arts and Sciences and the National Science Foundation (Grant #SBR-9319835) for funding support for
this project, and the Economics Science Laboratory at the University of Arizona for making
available their software for conducting normal form game experiments, and Rachel Croson
and an anonymous reviewer for useful suggestions.
∗∗ Send inquiries to: James Walker, also [email protected]
138
tion for the existence of government is that many of the dilemmas faced by
modern society take on the structure of a PD game. It has been argued that,
without government intervention to change payoffs through inducements or
fines, these public dilemmas will remain unsolved (Taylor, 1987).
Field research, however, has found that many individuals in everyday life
do not seem to follow the prescriptions of non-cooperative game theory. Many
individuals in fact contribute to public causes without being coerced, work
hard when they are on a team, and voluntarily refrain from polluting the environment. These apparent anomalies may be explained, however, by factors
that may change the payoff matrix faced by an individual from a PD game to
some other game. In the field it is hard to know whether individuals actually
face a PD game since there are many uncontrolled variables that may account
for observed behavior.
In contrast, experimental research allows for both a tight control of the
exact structure of the objective payoffs and for the introduction of carefully
designed changes in the experimental situation that can help assess which
variables may account for behavior. Experimental research of PD games
has shown that subjects frequently violate the dominant strategy prediction
(Davis and Holt, 1993: 94–96; Roth, 1995). These results have been replicated often and, thus, a serious effort needs to be made to provide an
explanation for why individuals would choose a dominated strategy. This
effort is particularly important for scholars of public choice as PD games
lie at the heart of many questions addressed in this field.
Theoretically, it has been shown that cooperation can occur in equilibrium
if the game is repeated infinitely. PD experiments, however, are typically
one-shot or, if repeated, do not last much more than an hour. Thus, this
infinitely repeated theory is not entirely helpful in an experimental setting. In
the context of a finitely repeated game, Kreps, Milgrom, Roberts, and Wilson
(1982) have shown that cooperation can occur in equilibrium if players have
sufficient priors that other players will respond cooperatively to cooperative
play. Given this, expectations about other players’ types may be important
in explaining cooperation in dilemma situations, but additional factors are
probably at play.
Contrary to standard game theoretical predictions, Rapoport (1967) and
Rapoport and Chammah (1965) argued that cooperation in repeated PD
games depends, among other things, on the cardinality of the relationships
among the game payoffs as well as their overall structure (or ordinal relationships). Let us illustrate their argument with the two-person, PD game
that is shown in Figure 1 using the concepts assigned by Rapoport and
Chammah (1965) to each of the payoff entries based on their substantive
meanings: Temptation (T), Reward (R), Punishment (P), and Sucker (S)
139
Figure 1. Interpreting PD game payoffs.
where T > R > P > S. There are two kinds of pressure to defect. First,
if the other player were to cooperate, a player can increase his or her own
payoff by defecting. This gain in payoffs (T–R) is referred to as Greed. Similarly, if the other player were to defect, cooperating costs (P–S). This loss in
payoffs is referred to as Fear. There is an incentive at the level of the group,
however, to cooperate, because both players are worse off than if they both
defect. The magnitude of this incentive is (R–P), referred to as “Cooperators’
Gain”. Social psychologists have argued that players will pay attention to the
size of Greed, of Fear, and of Cooperators Gain in making their decisions
while these cardinal values do not play a role in standard game theoretical
predictions (Komorita, Sweeney, and Kravitz, 1980; Bonacich, Shure, Kahan,
and Meeker, 1976).
In addition to the arguments about the importance of cardinal relationships, recent empirical evidence suggests that at least some subjects facing
a PD game in pecuniary benefits, interpret the game in utility space as if it
were a coordination game with two equilibria – both defect (D,D) and both
cooperate (C,C).1 Support for this view is based on evidence from questionnaire data. Faced with a set of questions that ask subjects to rank the four
outcomes of the game (where their strategy choice is shown first), many subjects rank the outcome (C,C) above the outcome (D,C) which is the reverse
of the expected ranking. For example, in an experimental investigation of a
one-shot PD game, using double blind procedures, Ahn, Ostrom, and Walker
(1999) report that 40% of a pool of 136 subjects revealed a preference for
(C,C) over (D,C) in their responses, and 27% were indifferent between the
two outcomes. The questionnaire was administered after subjects made their
game decision, but before subjects learned of the decision of the person with
whom they were matched in the PD game. The same question was asked of
181 undergraduates at Indiana University during the spring of 1999 and 27%
ranked the outcome (C,C) over (D,C), and 25% were indifferent.
A possible theoretical foundation for such preference rankings is given in
a model based on payoff inequity aversion developed by Fehr and Schmidt
(forthcoming). In this model, in addition to one’s own earnings, individuals’
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utility functions incorporate negative components based on relative payoffs
in the game. In particular, some types of individuals are assumed to have an
aversion to payoff outcomes in which their payoff outcome is below (above)
the payoffs of other players in the game. The strengths of these components
are assumed to vary across individuals and with respect to whether an individual’s payoff is above or below that of others.2 If an individual’s inequity
aversion is of sufficient magnitude and the individual believes that a sufficient
proportion of other players also have such preferences, the individual may
treat the PD game in pecuniary benefits as a coordination game in utility
space. In this coordination game, players face the problem of coordinating on
the payoff dominant equilibrium that corresponds to (C,C) in the PD game.3
For types of players who are averse to inequalities, the lower the value of
Greed, the higher the proportion of individuals who may see the game as a
coordination game rather than a PD game for any given distribution of types.
In the research reported here, we investigate the frequency of cooperative
play in four non-repeated PD games as a function of the payoff structure of
these games and the history of prior play in a series of coordination games.
Levels of Fear and Greed represented in the payoff structure are systematically varied to allow us to examine their impact on cooperative play. In
addition, the arguments above suggest that some individuals may treat these
PD games as coordination games. With this in mind, we investigate the conjecture that prior experience in coordination games will affect play in the PD
games.
The paper is organized as follows. In Section 2, we present the principal
elements of the PD games and our research hypotheses. Section 3 presents
the principal elements of the coordination games played by the subjects prior
to play of the PD games, as well as the details of the experimental protocols
that allow for investigating the role of social history and reputation building.
Section 4 presents key elements of the experimental decision environment.
Results are presented in Section 5. Section 6 presents summary observations.
2. PD games: Payoff parameters
The four PD games investigated are shown in Figure 2. Table 1 associates
the game payoffs with the concepts of Fear, Greed, and Cooperators Gain,
as well as to concepts related to maximum and minimum payoffs for an
individual, and normalized values of Fear and Greed. Variations in Fear
(P–S) and Greed (T–R) are accomplished by varying T and S.4 We propose
the following two behavioral conjectures:
Conjecture 1: The rate of cooperation will be decreasing in Fear (P–S).
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Figure 2. PD game structures.
Conjecture 2: The rate of cooperation will be decreasing in Greed (T–R).
These conjectures generate five research hypotheses related to the expected
frequency of cooperative choices across pairs of games. Each hypothesis is
stated in terms of expected frequency of play of strategy C.
H-1FEAR : Game 1 > Game 2
H-2FEAR : Game 3 > Game 4
H-3GREED: Game 1 > Game 3
H-4GREED: Game 2 > Game 4
H-5FEAR AND GREED:
Game 1 > Game 4
Fear (P–S) is 10 in Game 1 and 40 in Game 2.
Fear (P–S) is 10 in Game 3 and 40 in Game 4.
Greed (T–R) is 10 in Game 1 and 40 in Game 3.
Greed (T–R) is 10 in Game 2 and 40 in Game 4.
Fear (P–S) and Greed (T–R) are 10 in Game 1
and 40 in Game 4.
3. Social history and reputation building
As noted above, a popular explanation for cooperative play in finitely repeated PD games is the reputation model of Kreps et al. (1982), in which
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Table 1. Interpreting payoff parameters in PD games
PD
game
1
2
3
4
Fear
Greed
Cooperator’s
gain
(Maximum –
minimum)
payoff
Fear
(normalized)
Greed
(normalized)
(P–S)
(T–R)
(R–P)
(T–S)
(P−S)
(T−S)
(T−R)
(T−S)
10
40
10
40
10
10
40
40
40
40
40
40
60
90
90
120
1/6
4/9
1/9
1/3
1/6
1/9
4/9
1/3
cooperation by an individual can be justified by the belief that other players have incentives other than those described by the game. An important
component of this model is that players use previous play of the game to
update their beliefs. In related empirical research, Knez and Camerer (1996)
consider the possibility that players can establish reputations that can signal a
player’s likely behavior across alternative games. In particular, they show that
the precedent of efficiency in n-player coordination games increases the level
of cooperative play in subsequent repeated n-player social dilemma games.
Cain (1998) investigates a design in which prior play in a dictator game
is related to own behavior in a PD game and knowledge of another’s play
in a dictator game affects expectations of that person’s play in a PD game.
Schotter (1998) examines a two-part design. In Part I, subjects play a trustinducing game. The subjects play a minimum-effort coordination game to
induce “low trust” and a median-effort coordination game to induce “high
trust”. In Part II, subjects play a profit sharing game in which players face
the challenge of coordinating on a high-pay equilibrium versus a less risky
low-pay equilibrium.
If some subjects view the PD game as if it were a coordination game, it
seems reasonable to conjecture that these players will base their play in this
game on their beliefs that others also view it as a coordination game and their
beliefs about how others will play this coordination game. As discussed in the
introduction, the first type of belief can be affected by the level of Greed in
the games. Experimental evidence suggests that the second type of belief is
influenced by prior play of coordination games. In particular, the frequency of
play of a payoff dominant equilibrium has been found to be dependent upon
the history of prior play (Schmidt, Shupp, Walker, and Ostrom, 1998).
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Figure 3. Coordination game structures.
In the research reported here, the history of play in coordination games
is examined for its impact on the rate of cooperation in non-repeated PD
games. The experimental sessions consisted of two phases. In Phase I, subjects played one of the coordination games displayed in Figure 3 for eight
rounds of play, with feedback after each round. In Phase II, each subject made
a decision in each of the four PD games shown in Figure 2 and in each of
the coordination games shown in Figure 3, without feedback until all games
were completed. Each of the four coordination games shown in Figure 3 has
two pure strategy Nash equilibria, (A,A) and (B,B). In all four games, the
equilibrium (B,B) is payoff dominant. The four games differ, however, in the
differences in payoffs between (B,B) and (A,A) and in the risk in foregone
payoffs of playing strategy B over strategy A.5
The Phase I coordination games were played utilizing two matching protocols, random matching and fixed matching. The matching protocols were
announced publicly. In the “random matching protocol”, subjects were randomly matched, without replacement, each decision round. That is, they were
never matched with the same person twice. After each decision, subjects
observed the outcome of their own action and the action of the individual
with whom they were matched. In this protocol, subjects gained historical
information related to the play of subjects from the pool of subjects with
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whom they were playing, but had no opportunity to build reputations through
repeated play with the same person.6 We refer to this condition as one of
“social history”. In the “fixed matching protocol”, however, subjects were
randomly matched before the first game decision, and subsequently made all
decisions matched with that same person. We refer to this condition as one
of “reputation building”. If history has no substantive effect on behavior, we
should observe no significant differences in play of the PD games across the
two matching protocols.
4. Experimental setting
Subjects were paid volunteers from undergraduate courses in economics at
Indiana University.7 Prior to volunteering, subjects were informed that they
would participate in a decision-making exercise and would be paid in cash
an amount dependent upon their decisions and the decisions of others in the
experiment. Subjects were recruited in cohorts of size 10. The experimental
design was balanced. In total, there were 32 sessions, 16 utilizing the random
matching protocol and 16 utilizing the fixed matching protocol. Since there
were 10 subjects making decisions in each session, there are a total of 320
decisions for each PD game, 160 under the conditions of each of the two
protocols.
Upon arriving at the experimental laboratory, subjects were randomly
seated at computer monitors. They were informed that the experimental session would consist of two phases and at the end of the experimental session
they would be privately paid their earnings in cash, plus $5 for keeping
their appointment. Subjects were instructed to think of the payoffs in the
individual games as “computer pesos”, where the conversion rate was 100
pesos equals $1. Subjects, on average, earned $15 to $20 and participated for
approximately 45 minutes.
The games played were described as board games with a row and column
player. Each subject made a choice between option “D” and option “C”. In
the experiment the rows were actually labeled “A” and “B”. Subjects were
given complete information about the payoff structure. Each player always
saw himself/herself as a row player. Before beginning Phase I, subjects were
informed publicly of the matching protocol to be used throughout the game
(random or fixed as described above). After each decision round in Phase I,
each subject was informed of his/her payoff and the decision and payoff of the
subject with whom he/she was matched. Subjects did not receive information
on the decisions of other pairs of subjects.
Upon completion of Phase I, subjects received a handout describing Phase
II. The handout explained they would participate in a final group of decision
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games. In experiments using the random matching protocol, subjects were
informed that they would be matched with a person with whom they were
never matched in Phase I. In experiments using the fixed matching protocol,
subjects were informed that they would be matched with the same person
with whom they were matched in Phase I. The subjects were presented with
a game sheet that contained each of the four 2 × 2 symmetric bi-matrix PD
games. Upon completing their selection for all four games, their sheets were
collected and their earnings were calculated.
5. Results
5.1. Cooperative choices across games and protocols
The top panel of Table 2 reports the frequencies of cooperation across protocols and game structures. Cooperation is chosen on average 32% in the
random matching protocol and 42% in the fixed matching protocol. Table 2
also reveals considerable variation in the level of cooperation: Game 1 (low
fear and low greed) in the fixed matching protocol generated the highest rate
of cooperation (47%), while Game 3 (low fear and high greed) in the random
matching protocol generated the lowest (26%).8
Beyond these pooled results, one can also examine the distribution of
cooperative choices across individuals. The lower panel of Table 2 displays
summary information about the percentage of individuals who never made
a cooperative choice C in any of the PD games they played, chose C once,
chose C twice, chose C three times, or chose C in all four PD games. The
data suggests substantial variation in strategy choices across individuals, with
46% of all subjects never choosing to cooperate and 23% always choosing
to cooperate. The other interesting observation is twice as many individuals
chose to cooperate in all four PD games in the fixed matching protocol in
contrast to the random matching protocol.
5.2. Research hypothesis based on fear and greed
Table 3 reports the results of statistical tests of the five research hypotheses
based on the values of Fear and Greed. For each paired game comparison, the
frequency of cooperative choices is displayed, as well as χ 2 test statistics for
each research hypothesis for each of the matching protocols. In general, there
is not a consistent pattern of support found for the research hypotheses across
the concepts of Fear and Greed or across the two matching protocols. Only
Hypothesis H-3, based on the value of Greed (Greed is 40 in Game 3, 10 in
Game 1), is supported in both protocols. Inconsistent with H-2, the subjects
cooperated more frequently in Game 4 than in Game 3.
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Table 2. Frequency of cooperative choices in PD games
Pooled frequencies of cooperative choices (C) across game structures
Game 1
Random protocol 65 (41%)
Fixed protocol
75 (47%)
Total
140 (44%)
Game 2
Game 3
Game 4
Total
49 (31%)
69 (43%)
118 (37%)
41 (26%)
61 (38%)
102 (32%)
47 (30%)
66 (41%)
113 (35%)
202 (32%)
271 (42%)
473 (37%)
160 decisions in each cell: 10 subjects × 4 games × 4 experimental sessions.
640 decisions for each protocol: 160 × 4.
320 decisions for each game structure.
Distribution of cooperative choices (C) across individuals
Individuals
Individuals Individuals
Individuals
Individuals
with 0
with 1
with 2
with 3
with 4
choices of C choice of C choices of C choices of C choices of C
Random protocol 79 (49%)
Fixed protocol
67 (42%)
Total
146 (46%)
21 (13%)
22 (14%)
43 (13%)
23 (14%)
13 (8%)
36 (11%)
13 (8%)
9 (6%)
22 (7%)
24 (15%)
49 (31%)
73 (23%)
The numbers in the first row represent the numbers of cooperative choices (C). 79 (49%) in
the first column of the second row means that 79 subjects (49% of the total of 160 subjects)
in the Random Matching Protocol never made any C choices among the four PD games they
faced.
To this point, only aggregate data has been used to test the effects of Fear
and Greed on frequency of cooperative choices. An alternative test of the research hypotheses can be made by examining each individual’s decisions for
various pairs of game structures. As an alternative formulation of the research
hypotheses, the hypotheses can be restated as they would relate to the play of
an individual across pairs of the four games. For each game, an individual
must choose strategy C or D. Thus, for each pair of games, an individual
is implicitly choosing one of four possible pairs of action, {D in one game,
D in the other game}, similarly {D, C}, {D, D}, or {C, D}. For each research
hypothesis, only one of these observations is a violation. For example, H1: Game 1 > Game 2 can be reinterpreted as an expectation of play for an
individual across Games 1 and 2. The choice of D in Game 1 and C in Game
2 would be a violation of this hypothesis. Behavior that would be consistent
with this hypothesis would be the play of C in Game 1 and D in Game 2, the
play of D in both games, or the play of C in both games. Note that if subjects
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Table 3. Hypotheses tests – frequency of cooperative choices in PD games
Research
hypotheses
Basis for
hypotheses
Random match
protocol
Fixed match
protocol
H-1: Game 1 > Game 2
Fear
H-2: Game 3 > Game 4
Fear
H-3: Game 1 > Game 3
Greed
H-4: Game 2 > Game 4
Greed
H-5: Game 1 > Game 4
Fear & Greed
C in Game 1 = 65
C in Game 2 = 49
χ 2 = 3.4883∗
C in Game 3 = 41
C in Game 4 = 47
No test
C in Game 1 = 65
C in Game 3 = 41
χ 2 = 8.1256∗∗∗
C in Game 2 = 49
C in Game 4 = 47
χ 2 = 0.0592
C in Game 1 = 65
C in Game 4 = 47
χ 2 = 4.4505∗∗
C in Game 1 = 75
C in Game 2 = 69
χ 2 = 0.4545
C in Game 3 = 61
C in Game 4 = 66
No test
C in Game 1 = 75
C in Game 3 = 61
χ 2 = 2.5064∗
C in Game 2 = 69
C in Game 4 = 66
χ 2 = 0.1153
C in Game 1 = 75
C in Game 4 = 66
χ 2 = 1.0270
∗ = p < .1
∗∗ = p < .05
∗∗∗ = p < .01
Table 4. Frequency of paired game violations by individuals
Research
hypotheses
Basis for
hypotheses
Random match
protocol
Fixed match
protocol
H1: Game 1 > Game 2
H2: Game 3 > Game 4
H3: Game 1 > Game 3
H4: Game 2 > Game 4
H5: Game 1 > Game 4
Fear
Fear
Greed
Greed
Fear & Greed
3.75%
10.63%
4.38%
7.5%
6.88%
3.75%
10.00%
4.38%
5.63%
5.00%
were making decisions randomly, one would expect violations on average
25% of the time since there are four possible choice combinations. Table 4
reports the frequency of violations for each research hypothesis. For all five
hypotheses, the frequency of violations is smaller than 25%. The frequency
of violations is greatest for the research hypothesis based on Fear for Games 3
and 4, consistent with the aggregate results reported in Table 3 for the paired
comparisons on those games.
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5.3. Combining prior experience in coordination games with fear and greed
The paired comparisons reported above do not control for the history of play
that subjects encountered prior to their strategy choices in the PD games. The
experimental design allows one to examine how the precedent of choices in
coordination games played in Phase I of the design affects strategy choice
in the PD games of Phase II. It is well known that players often cooperate
in one-shot PD games, so we seek to determine if their past experiences in
non-PD games help to explain the frequency of cooperation. The behavioral
conjecture is that subjects who experienced a greater frequency of play of the
payoff dominant strategy (B) in the coordination games of Phase I are more
likely to follow the cooperative strategy in the PD games of Phase II.
Figure 4 plots the frequency of play of cooperative choices conditional
on the number of times an individual encountered the play of the payoff
dominant equilibrium in the coordination game they played in Phase I in
the experimental sessions. That is, as shown in the top panel for the games
using the random matching protocol, the frequency of cooperative choices
is plotted for those subjects who observed B (the payoff dominant strategy)
being played zero times by one of the players with whom they were matched
(2 subjects), those who observed B being played 1 time (4 subjects), . . . up to
those who observed the play of B in all 8 decision rounds in Phase I (15 subjects). The bottom panel of Figure 4 plots the same information, but for those
experimental sessions utilizing the fixed matching protocol. As shown, these
histograms suggest a much stronger effect of history in the fixed matching
protocol than in the random matching protocol.
Using a logit regression model, we can more formally integrate the history
of play as an explanatory variable in predicting the rate of cooperation across
PD games. The history of play, through its impact on social history (random
matching protocol), is examined separately from its impact on reputation
building (fixed matching protocol). Table 5 presents results from the logit
model. The absolute values of the payoff parameters for Fear and Greed are
included as explanatory variables. The variable History records the number
of times a subject encountered play of B by the persons/person with whom
they were matched in the coordination games of Phase I. In addition, dummy
variables are included (Coord-Game 2, Coord-Game 3, and Coord-Game 4)
to control for the particular coordination game subjects played in Phase I,
and for whether a subject was experienced (they had previously participated
in one of the experimental sessions as inexperienced subjects).9 In addition to
the logit model coefficients, p-values for a two-sided t-test on the significance
of the coefficients are reported.10
The baseline observation in this regression is the action taken by an inexperienced subject who played coordination Game 1 in Phase I. The primary
149
Figure 4. Play of C in PD games as a function of history in coordination games.
results from this logistic regression can be summarized as follows. Consistent
with the results found in the paired game comparisons, support for Fear and
Greed as explanatory variables is rather weak. The coefficients are statistically significant only in the case of Greed in the random matching protocol.
On the other hand, the variable History is statistically significant in both
protocols. Further, as expected, the impact of history is much larger in the
case of the fixed matching protocol (.287) relative to the random matching
protocol (.173). As an example of the magnitude of the effect of history in
the fixed matching protocol, consider an individual from the baseline situation
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Table 5. Logit model – history and absolute values of fear and greed
Dependent variable: Probability of C.
Random match protocol
Coefficient
p-value
Constant
Fear
Greed
History
Coord-Game 2
Coord-Game 3
Coord-Game 4
Experience
–1.255
–0.005
–0.013
0.173
0.672
–0.281
0.108
–0.648
0.002
0.381
0.023
0.001
0.005
0.271
0.699
0.003
Fixed match protocol
Coefficient
p-value
–1.580
–0.001
–0.008
0.287
0.052
–0.339
–0.259
–0.609
0.000
0.932
0.149
0.000
0.831
0.158
0.286
0.003
and playing PD Game 1 (low Fear and low Greed). The model predicts that
a subject with a history of having encountered no instances of the payoff
dominant strategy in Phase I would choose cooperation with a probability of
16%. On the other hand, if that individual encountered the payoff dominant
strategy in every game they played in Phase I (8 plays), the model predicts a
probability of cooperation of 65%.11
5.4. An alternative specification of fear and greed
The design of the PD games investigated in this study called for a systematic
variation in Fear (P–S) and Greed (T–R), while holding constant the “cooperators gain” (R–P). One consequence of this approach is that the values
of (T–S), the difference between the minimum and maximum payoffs for
a game, vary across the game structures (see Table 1). In this section, an
alternative measurement for Fear and Greed is proposed in which these variables are “normalized” for the variation in (T–S) across the PD games. This
approach provides for an interesting link to the approach taken by Anatol
Rapoport.
In Rapoport (1967), the rate of cooperation, K, is represented as a function
of the payoff entries
K = K(T, R, P, S).
(1)
151
Various functional forms of K have been examined in a search for an
index of cooperation for the Prisoner’s Dilemma. The simplest and most well
known among them is
R−P
(2)
T−S
Note that (2) is simply cooperators’ gain, normalized for T–S. When both
Fear and Greed are normalized along with the cooperators’ gain, we find that
the sum of the three normalized values is always 1.
K=
R−P T−R P−S
T−S
+
+
=
=1
(3)
T−S
T−S
T−S
T−S
Substantively, (3) implies the pressures to defect (Fear and Greed) and
the pressure to cooperate (Cooperators’ Gain) contend with each other. The
functional form of K in (2), therefore, can be expressed as 1 minus the sum
of the two pressures to defect
T−R P−S
−
(4)
T−S
T−S
Equation (4) includes implicit assumptions that: 1) when there are no gains
from cooperation (R=P), players always defect (K=0), and 2) when there
is neither Fear nor Greed (T=R and P=S), players always cooperate (K=1).
Some experimental studies, however, have shown behavioral inconsistencies
with these assumptions (Scodel, Minas, Ratoosh, and Lipetz, 1959; Minas,
Scodel, Marlowe, and Rawson, 1960; Rapoport, 1967; Sajio and Nakamura,
1995; Palfrey and Prisbrey, 1997).
We propose an alternative specification that allows K to vary so as to allow
for such behavior.
K= 1−
T−R
P−S
+γ
(5)
T−S
T−S
where it is assumed that 0 < α < 1, −1 < β < 0 and −1 < γ < 0. Note that
when α = 1, β = −1, and γ = −1, (5) is equivalent to (4). The normalized
values of Fear and Greed, (T–R)/(T–S) and (P–S)/(T–S), will be denoted as
Fear(N) and Greed(N). The values of Fear(N) and Greed(N) are reported in
Table 1. Note that α in (5) can be interpreted as the propensity to cooperate
in the absence of Fear and Greed, while β and γ are the weights attached to
the two pressures to defect.
The magnitude and significance of these parameters are statistically tested
with a logit model in which other control variables are also included, as
in Table 5.12 Table 6 reports the results from this alternative specification
in the random matching protocol and fixed matching protocol. The primary
K=α+β
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Table 6. Logit model – history and normalized values of fear and greed
Dependent variable: Probability of C.
Independent
Random matching
variable
Coefficient
p-value
Constant
Fear(N)
Greed(N)
History
Coord-Game 2
Coord-Game 3
Coord-Game 4
Experience
–0.743
–1.479
–2.227
0.173
0.675
–0.282
0.108
–0.650
0.147
0.066
0.006
0.001
0.005
0.270
0.699
0.003
Fixed matching
Coefficient
–1.329
–0.541
–1.236
0.287
0.052
–0.339
–0.260
–0.609
p-value
0.005
0.493
0.118
0.000
0.831
0.158
0.285
0.003
new finding is the significant impact of the normalized versions of Fear and
Greed in games played in the random matching protocol. Further, the results
suggest that game parameters play a more important role in PD games played
in the random matching protocol and that History plays the greater role in
games played in the fixed matching protocol. In addition, the coefficients for
Greed(N) are uniformly larger (in absolute value) than for Fear(N), suggesting a larger behavioral impact from variations in normalized measurements
of Greed relative to Fear.
6. Conclusions
Understanding the behavioral foundations of cooperation in social dilemma
games has been a core issue in the behavioral sciences for generations. Its
relevance to social and economic relationships is far-reaching (see Ostrom,
1998). Generally, dilemma situations in the field are nested in broad and complex relationships among decision makers. Understanding how these interrelationships affect cooperation is essential to building a broader understanding
of cooperation and trust in all social and economic interactions.
The research reported in this study extends this line of inquiry in two
fundamental ways. Payoff parameters are systemically varied to allow for
further testing of the relationship between Fear (the loss in payoff relative to
the dominant strategy payoff a player receives from cooperating if the other
player defects) and Greed (the gain in payoff relative to the Pareto optimum
payoff a player receives from defection if the other player cooperates). In
153
addition to examining the effects of changes in the absolute values of Fear
and Greed, new concepts related to “normalized” measurements of these variables are analyzed. The experimental design also extends previous analyses
by examining how the history of play from prior interactions in coordination
games affects the decision to cooperate. By examining two distinct matching
protocols, the results provide a perspective for beginning to understand the
impact of information as it relates to history of play in both a reputation
building and social history context.
Overall, the results failed to support the proposition that absolute values
of Fear and Greed are consistent predictors of behavior. Normalized values
of these variables, however, are found to be good predictors of behavior
within the context of the random matching protocol. In addition, normalized
measurements of Greed were found to have a larger impact on behavior than
normalized values of Fear.
Knez and Camerer (1996) found that the precedent of efficient play in
coordination games facilitated more cooperative play in a finitely repeated
social dilemma game. The results reported here support those findings. In
both the reputation and social history contexts, history of prior play in coordination games is a statistically significant predictor of cooperation in the
PD game environment. However, the importance of history is significantly
more pronounced if players are matched repeatedly with the same person
who can thus build a reputation, rather than never rematched with the same
person.
Consequently, we should expect to find higher levels of public goods
provided, less shirking, and a more sustainable use of resources in those
dilemma situations where participants interact repeatedly with the same set
of individuals and reputations can be built. On the other hand, as the relative
level of temptation increases in a dilemma situation one should observe a
decrease in the level of cooperation. Thus both the historical setting and
the cardinality of payoff parameters affect the likelihood of cooperation in
dilemma situations.
Notes
1. See Hayashi, Ostrom, Walker, and Yamagishi (1999) for a discussion of the literature
related to this view.
2. An alternative explanation might be found in the work of Palfrey and Prisbrey (1997).
They specify a utility function, in the context of voluntary contributions experiments, that
includes the warm-glow term and other subjects’ payoffs as well as one’s own monetary
payoff.
3. See Ahn, Ostrom, and Walker (1999) for further elaboration.
154
4. Rapoport and Chammah (1965) conducted a similarly designed experiment using seven
PD game matrices. Subjects in their experiment played PD games three hundred consecutive times in one of five experimental conditions. Across three matrices, T(S) varied from
2 to 10 to 50 while R and P remained constant at 1 and –1. The average cooperation rate
was 66% (T=S=2), 46% (T=S=10), and 27% (T=S=50), supporting hypothesis H-5 in the
text.
5. See Schmidt et al. (1998) for a more extensive discussion of the theoretical properties of
these games, as well as observed play.
6. Berg, Dickhaut, and McCabe (1995) investigate the role of social history in a “trust game”
where subjects see the history of play from a previous experimental session in which they
did not participate. Thus, we investigate social history in the sense of a smaller sample,
but from the perspective of a more immediate population.
7. These subjects were generally not economics majors. Introductory economics classes at
Indiana University enroll students with majors across numerous disciplines. Alternates
were recruited in case someone did not make the scheduled experimental session.
8. Ahn et al. (1999) report results from play in the same four PD games in a one-shot setting,
with no prior play among the subjects. In that study they find the following rates of
cooperation, Game 1 (27.5%), Game 2 (30%), Game 3 (30%), Game 4 (21.5%).
9. There were four sessions for each of the four coordination games in Phase I. One among
the four sessions for each game structure used a group of subjects recruited from the pool
of subjects who participated in one of the previous sessions using the Random Matching
Protocol.
10. We have also conducted OLS analyses with the same specification of independent variables. The results were virtually identical in terms of the significance of the coefficients.
11. One might argue for investigating interaction effects between history and fear and greed,
respectively. If history of prior play leads a player to think the other player is likely to
defect, it seems that fear would be more important than greed. Conversely, if a player
thinks the other player is likely to cooperate, it seems greed would be more important
than fear. These interaction terms, however, were not found to be statistically significant.
12. Using a standard linear regression model does not affect the qualitative conclusions of our
analysis.
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