Equilibrium Vengeance
Daniel Friedman and Nirvikar Singh
September 2001
Abstract
This paper introduces two ideas, emotional state dependent utility components (ESDUCs), and evolutionary perfect Bayesian equilibrium (EPBE).
Using a simple extensive form game, we illustrate the e¢ciency-enhancing
role of a powerful ESDUC, the vengeance motive. Incorporating behavioral noise and observational noise leads to a range of (short run) Perfect
Bayesian equilibria (PBE) involving both vengeful and non-vengeful types.
We then derive a (long run) non-trivial EPBE where both types survive and
reap mutual gains, as well as a trivial EPBE with only the non-vengeful
type and no mutual gains.
This is an incomplete …rst draft. Please ask permission before circulating further. Comments of all sorts are very welcome; email [email protected].
1.
Introduction
This paper makes two contributions. First, it spotlights a puzzle
regarding human behavior, and o¤ers a possible solution. The behavior in question is vengeance: when some culprit harms you or your
loved ones, you may choose incur a substantial personal cost to harm
him in return.
A taste for vengeance, the desire to ”get even,” is so much a part of
daily life (and the evening news) that it is easy to miss the evolutionary
puzzle. We shall argue that indulging your taste for vengeance in
general reduces your material payo¤ or …tness. Absent countervailing
forces, the meek (less vengeful people) should have inherited the earth
long ago, because they had higher …tness. Why then does vengeance
persist?
The present paper answers the puzzle by showing how in equilibrium vengeful individuals can achieve equal …tness with the less
vengeful in a world with behavioral and observational noise.
The second contribution is an apparently new equilibrium concept, evolutionary perfect Bayes equilibrium (EPBE). We believe that
EPBE is germane in a wide variety of applications. It de…nes long
run equilibrium (generalizing the equal pro…t condition of competitive
Equilibrium Vengeance
2
markets) for games of incomplete information with possible entry, exit
and/or switching among multiple player types. In this paper we use
EPBE to show how vengeance can persist despite its apparent …tness
handicap.
Vengeance is closely tied to several vexing issues, methodological
and substantive. To clear the underbrush, we begin with preliminary
discussions on the nature of social dilemmas, the meaning of positive and negative reciprocity, evidence on why both are important to
economists, and various modeling approaches employed so far. Section
2 presents a simple extensive form model of the basic dilemma, and
shows how the equilibrium changes once vengeance is included. Section 3 introduces the viability problem that arises when an individual’s
vengefulness can’t be perfectly known in advance and when behavioral
errors are possible. Section 4 derives three short-run perfect Bayes
equilibria, two pooling equilibria and one separating equilibrium. Section 5 looks at the long-run in which the nature and prevalence of types
can evolve, and derives the resulting EPBE. Following a concluding
discussion, Appendix A collects the mathematical details.
2.
Preliminaries
Figure 1 elucidates the fundamental social dilemma in terms of
net material bene…t (x > 0) or cost (x < 0) to ”Self” and bene…t or
cost (y >0 or <0) to counterparties, denoted ”Other”. Economists
think most often about the mutual gains quadrant I, where actions
simultaneously bene…t Self and Other. Such symbiotic actions increase
social e¢ciency.
Quadrant IV is the well-studied opportunistic region, where Self
bene…ts at Other’s expense; the biological terms are parasitism and
predation. The ‡ip side is the altruism quadrant II, where Self bears
a personal cost in order to bene…t Other. Cipolla (1976) refers to
behavior producing outcomes in quadrant III as stupidity, but we shall
explain it later as vengeance.
Social dilemmas arise from the fact that evolution directly sup-
Equilibrium Vengeance
3
ports behavior that bene…ts Self, i.e., outcomes x > 0 in quadrants IV
(or I) but not x < 0 in II (or III), while in contrast, e¢ciency requires
outcomes above the diagonal [x + y = 0]. Social creatures (such as
humans) thrive on cooperation, by which we mean devices that support outcomes in II+ and discourage outcomes in IV-. Such devices
somehow internalize Other’s costs and bene…ts.
Other’s
benefit
II. Altruism
e.g., favors,
parental investment
y
r = 1/8
I. Mutual Gains
e.g., market
exchange, symbiosis
D
II+
C
Social
gain
II-
x
Self’s
cost
O
Social
loss
IV+
IV-
IV. Opportunism
III. Vengeance
e.g., shirking,
predation and parasitism
e.g., strikes, feuds
Other’s
cost
Figure 1: Payo¤s to Self and Other
Self’s
benefit
Equilibrium Vengeance
4
2.1. How Can Cooperation Persist?
Biologists emphasize the device of genetic relatedness. If Other is
related to Self to degree r>0, then a positive fraction other’s payo¤s
are internalized via ”inclusive …tness” (Hamilton, 1964) and evolution
favors outcomes above the line [x + ry = 0]. For example, the unusual
genetics of insect order hymenoptera lead to r = 3=4 between sisters,
so it is no surprise that most social insects (including ants and bees)
belong to this order and that the workers are sisters. For humans
and most other species, r is only 21 for full siblings and for parent and
child, is 1/8 for …rst cousins, and goes to zero exponentially for more
distant relations. On average r is rather small in human interactions,
as in the steep dashed line in Figure 1, since we typically have only
a few children but work and live in groups with dozens of individuals. Clearly non-genetic devices are needed to support human social
behavior.
Economists emphasize devices based on repeated interaction, as
in the ”folk theorem” (cites). Suppose that Other returns the bene…t (”positive reciprocity”) with probability and delay summarized in
discount factor ± 2 [0, 1). Then that fraction of other’s payo¤s are
internalized (Trivers, 1971) and evolution favors behavior producing
outcomes above the line [x + ±y = 0]. This device can support a large
portion of socially e¢cient behavior when ± is close to 1, i.e., when
interactions between two individuals are symmetric, predictable and
frequent. But humans specialize in exploiting once-o¤ opportunities
with a variety of di¤erent partners, and here ± is small, as in the same
steep dashed line. Other devices are needed to explain such behavior.
Here we will emphasize devices based on other-regarding preferences. For example, suppose Self gets a utility increment of ry. Hence
Self partially internalizes the material externality, and undertakes behavior that is above the line [x + ry = 0]. Friendly preferences, r 2 [0,
1], thus can explain the same range of behavior as genetic relatedness
and repeated interaction. However, by itself the friendly preference
device is evolutionarily unstable: those with lower positive r will tend
Equilibrium Vengeance
5
to make more personally advantageous choices, gain higher material
payo¤ (or …tness), and displace the more friendly types. Friendly
preferences therefore require the support of other devices.
Vengeful preferences rescue friendly preferences. Self’s material
incentive to reduce r disappears when others base their values of r
on Self’s previous behavior and employ r < 0 if Self is insu¢ciently
friendly. Such visits to quadrant III will reduce the …tness of less
friendly behavior and thus boost friendly behavior. But visits to quadrant III are also costly to the avenger, so less vengeful preferences
seem …tter. What then supports vengeful preferences: who guards the
guardians? This is the central question in the present paper.
2.2.
How to model other regarding preferences?
Two main approaches can be distinguished in the recent literature.
The distributional preferences approach is exempli…ed in the Fehr and
Schmidt (1999) inequality aversion model, the Bolton and Ockenfels
(1999) mean preferring model, and the Charness and Rabin (1999)
social maximin model. These models begin with a standard sel…sh
utility function and add additional terms capturing self’s response to
how own payo¤ compares to other’s payo¤s. In Fehr-Schmidt, for
example, my utility decreases (increases) linearly in your payo¤ when
it is above (below) my own payo¤.
The other main approach is to model reciprocal preferences directly. Building on the Geanakoplos, Pearce and Stacchetti (1989)
model of psychological games, Rabin (1993) constructs a model of
reciprocation for two player normal form games, extended by Dufwenberg and Kirchsteiger (1998) and Falk and Fischbacher (1998) to somewhat more general settings. The basic idea is that my preferences
regarding your payo¤ depends on my beliefs about your intentions,
e.g., if I believe you tried to increase my payo¤ then I want to increase
yours. Such models are intractable except in the simplest settings.
Levine (1998) improves tractability by replacing beliefs about others’
intentions by estimates of others’ type.
Equilibrium Vengeance
6
We favor a further simpli…cation. Model reciprocal preferences
as state dependent: my attitude towards your payo¤s depends on
my state of mind, e.g., friendly or vengeful, and your behavior systematically alters my state of mind. This emotional state-dependent
other-regarding (ESDUC) approach is consistent with Sobel (2000)
and is hinted at in some other papers including Charness and Rabin.
The approach is quite ‡exible and tractable, but in general requires
a psychological theory of how states of mind change. Fortunately a
very simple rule will su¢ce for present purposes: you become vengeful towards those who betray your trust, and otherwise have standard
sel…sh preferences.
Empirical evidence is now accumulating that compares the various
approaches (Kagel and Wolfe, 1999; about 10 other cites). Cox (1999)
and some authors of the distributional models …nd evidence favoring
their models, but all other authors …nd evidence mainly favoring state
dependent or reciprocal models. Our own reading of the evidence
convinces us to focus on state dependent preferences, while noting
that distributional preferences may also play some role.
To formalize these ideas, the …rst step is to model explicitly the
underlying social dilemma.
3.
The Underlying Game
We use a simple extensive form version of the prisoner’s dilemma,
sometimes known as the Trust game (cite), shown in Panel A of Figure
2 (with payo¤s graphed in Figure 1). Player 1 (Self) can opt out (N)
and ensure zero payo¤s to both players. Alternatively Self can trust
(T) player 2 (Other) to cooperate (C), giving both a unit payo¤ and
a social gain of 2. However, Other’s payo¤ is maximized by defecting
(D), increasing his payo¤ to 2 but reducing Self’s payo¤ to -1 and
the social gain to 1. The basic game has a unique Nash equilibrium
found by backward induction (or iterated dominance): Self chooses
N because Other would choose D if given the opportunity, and social
gains are zero.
Equilibrium Vengeance
7
To this underlying game we add a punishment technology and a
punishment motive as shown in Panel B. Self now has the last move
and can in‡ict harm (payo¤ loss) h on Other at personal cost ch. The
marginal cost parameter c captures the technological opportunities for
punishing others.
F igure 2: Fitness Payoffs
A . Basic Trust G a m e
Self
O ther
T
D
(- 1, 2)
N
C
(0, 0)
(1, 1)
B . Extended Trust G a m e
Self
O ther
T
D
Self*
[ 0 ≤ h ≤ 8 ]
C
N
(0, 0)
(- 1 - ch, 2 - h)
(1, 1)
* U tility p a y o f f t o S e l f i s ln h - 1 - c h
C . Reduced Trust with a vengeance
Self
O ther
T
D
(- 1 - v, 2 - v/c)
C
N
(0, 0)
(1, 1)
Figure 2: Fitness Payo¤s
Self’s punishment motive is given by state dependent preferences.
If Other chooses D then Self receives a utility bonus of v ln h (but no
…tness bonus) from Other’s harm h. In other states utility is equal to
Equilibrium Vengeance
8
own payo¤. The motivational parameter v is subject to evolutionary
forces and is intended to capture an individual’s temperament, e.g., his
susceptibility to anger. See R. Frank (1988) for an extended discussion
of such traits. The functional forms for punishment technology and
motivation are convenient (we will see shortly that v parameterizes the
incurred cost) but are not necessary for the main results. The results
require only that the chosen harm and incurred cost are increasing in
v and have adequate range.
Using the notation ID to indicate the event ”Other chooses D,”
we write Self’s utility function as U = x + vID ln h; own material payo¤ x plus the relevant ESDUC. When facing a ”culprit” (ID
=1), Self chooses the reduction h in Other’s payo¤ so as to maximize
U = ¡1¡ch+v ln h. The unique solution of the …rst order condition is
h¤ = v=c and the incurred cost is indeed ch¤ = v. For the moment assume that Other correctly anticipates this choice. Then we obtain the
reduced game in Panel C. For sel…sh preferences (v = 0) it coincides
with the original version in Panel A with unique Nash equilibrium (N,
D) yielding the ine¢cient outcome (0, 0). For v > c, however, the
transformed game has a unique Nash equilibrium (T, C) yielding the
e¢cient outcome (1, 1). The threat of vengeance rationalizes Other’s
cooperation and Self’s trust.
3.1.
Can vengeful preferences evolve?
Consider evolution of the vengeance parameter v in an unstructured population. Assume for simplicity that the marginal punishment cost c is constant. Again for simplicity (and perhaps realism)
assume that, given the current distribution of v within the population,
behavior adjusts rapidly towards Nash equilibrium but that there is at
least a little bit of behavioral noise. Noise is present because equilibrium is not quite reached or just because the world is uncertain. For
example, Self may intend to choose N but may twist an ankle and …nd
himself depending on Other’s cooperative behavior, and Other may
intend to choose C but oversleeps or gets tied up in tra¢c. Such con-
Equilibrium Vengeance
9
siderations can be summarized in a behavioral noise amplitude e ¸ 0.
Also, Other may imperfectly observe Self’s true vengeance level v; say
that the observational noise amplitude is a ¸ 0.
The task now is to compute Self’s (expected) …tness W (v; a; e) for
each value of v at the relevant short run equilibrium, given the observational and behavioral noise. The function W de…nes a …tness landscape in which evolution pushes the evolving trait v uphill (Wright,
1949; Eshel, 198x; Kaufman, 1993).
First consider the case a = e = 0, where v is perfectly observed and
behavior is noiseless. Recall that in this case the short run equilibrium
(N, D) with payo¤ W = 0 prevails for v < c, and (T, C) with W =
1 prevails for v > c. Thus W (v; 0; 0) is the unit step function at
v = c. In the Appendix it is shown that with a little behavioral
noise (small e > 0) the step function slopes down, and with a little
observational noise (small a > 0) the sharp corners are rounded o¤,
as in Figure 3. As indicated, evolution pushes v downward towards 0
in the subpopulation initially below a level near c ¡ a, and pushes v
in the rest of the subpopulation to a level near c + a.
Equilibrium Vengeance
10
Figure 3: Fitness Was a Function of Vengefulness v
a=e=0
1
a, e > 0
0
a=0<e
_
v
c
Figure 3
4.
PBE
Figure 3 shows two local …tness maxima for Self, one at v=0 and
the other for v = vH > c;when a and e are both small and positive. The
Figure suggests that evolutionary forces will tend to move Selfs along
the …tness gradient, resulting in a fraction x of the Self population
with vengeance near vH and fraction (1 ¡ x) with vengeance near
x = 0: The fractions are respectively the fractions of the population
initially above and below the local minimum (near c ¡ a), and hence
are arbitrary.
This con…guration does not yet represent an equilibrium. One must
check that Other should attend to the signal; if the error amplitude a
is su¢ciently large relative to the fraction (1¡x) of non-vengeful types
Other would be better o¤ ignoring the signal and always playing C (see
Wittman, 1989, for an analogous situation in arms control.) Likewise,
Equilibrium Vengeance
11
if vengeful types are su¢ciently rare relative to the error amplitude,
Other might be better o¤ playing D irrespective of the signal.
These considerations are incorporated in perfect Bayes equilibrium
(PBE), suitably modifed to deal with large populations and trembles.
In PBE one arbitrarily and exogenously speci…es for each population
(Self and Other) the types and their relative prevalence and then requires that each type in each population picks the payo¤ maximizing
strategy (or …tness maximizing in the reduced game) in each subgame
given correct beliefs about the strategies chosen in the other population.
Nature
x(1-a)
xa
v ,1
(0,0)
(1,1)
(1-x)a
SH
(1-x)(1-a)
vH,0
(0,0)
S
O (1,1)
O
(-1-v, 2-vH/c)
0,0
(0,0)
O
(1,1)
S
(-1-v, 2-vH/c)
(-1, 2)
0,1
S
O
(-1, 2)
Figure 4: Game Tree
The game of incomplete information is shown in Figure 4. Nature
chooses among 4 possible Self types: true preferences with v = 0
or v = vH with probabilities x and 1 ¡ x; and realized signal s =
0 or s = 1 with symmetric error rate a = Pr[s = 0jv = vH ] =
Pr[s = 1jv = 0]: Self knows own preferences but not the realized signal,
and Other knows the realized signal but not the true type. Self’s
(0,0)
(1,1)
Equilibrium Vengeance
12
strategy (given his realized preference parameter v) is just the mixture
probability for chosing T; the unconstrained stategy set is [0; 1] but
with behavioral noise Self’s strategy set is [e; 1 ¡ e] for given 0 < e <
1=2: Other’s strategy is pair of mixture probabilities for choosing C
after observing respectively s = 0 and s = 1. The behavioral noise
constrained strategy set is [e; 1 ¡ e] £ [e; 1 ¡ e]: The payo¤s are as in
the reduced Trust game of Figure 2C.
We focus on the separating PBE in which the vH type of Self tries
to Trust (i.e., plays T with probability 1 ¡ e) and the v = 0 type
tries to Not Trust (i.e., plays T with probability e), while all Others
try to play C (and do so with probability 1 ¡ e) when the signal is
s = 1 and tries to play D when the signal is s = 0. Table 1 writes out
the resulting …tness outcomes and probabilities. Other’s behavior is
optimal as long as the expectation of the D payo¤ 2 ¡ v=c is greater
than (less than) the C payo¤ 1when he receives the signal s = 0 (resp.
s = 1). The condition simpli…es to E(vjs) · c (resp. ¸ c). The
probability that a s = 0 signal will arise in this separating PBE from
a vH type is x(1 ¡ e)a versus probability (1 ¡ x)e(1 ¡ a) from a v = 0
type. Routine algebra (or Bayes theorem) reveals that the critical
posterior expectation E(vjs = 0) = c corresponds to prior probability
a
(or population fraction) xs = 1=(1+( 1¡a
)( 1¡e
)( vHc¡c )). This condition
e
y
can be reexpressed in terms of log odds L(y) = ln( 1¡y
) as in the
proposition below. Note that L(y) is negative for 0 < y < 1=2, e.g.,
for reasonable values of y = a and e, so the inequalities may look a
bit strange.
Equilibrium Vengeance
13
PBE Probabilities
Fitness Payoff
Probability
Choice Self, Other
v = vH
v=0
(N, .) 0, 0
(T, C) 1, 1
(T, D) -(1+v), 2-v/c
(N, .)
(T, C)
(T, D)
0, 0
1, 1
-1, 2
Separating
Good Pooling
Bad Pooling
e
(1 – e)(1 – a)
(1 – e)a
e
(1 – e)(1 – e)
(1 – e)e
1-e
e.e
e(1 – e)
1- e
ea
e(1 – a)
e
(1 – e)(1 – e)
(1 – e)e
1-e
e.e
e(1 – e)
Notes:
In separating equilibrium, Other tries to play C if s = 1 and D if s = 0.
Other always tries to play C in good Pool, and always tries to play D in Bad Pooling.
The signal s = 1 with probability a 0 (0, ½) when v = 0 and is s = 0 with probability a when v
= vH.
The expression a = a(1 – e) + e(1 – a) = e + a – 2ae represents the probability that Other
chooses his less preferred action.
Figure 5: PBE Probabilities
Proposition 1 Given signalling technology with error amplitude a and
choice technology with tremble rate e, and given types v = 0 and
v = vH > c constituting respectively Self population fractions (1 ¡ x)
and x 2 (0; 1); assume that 0 < a; e < 1=2 and ® = a + e ¡ 2ae ·
1=(2 + vH ). Then the separating PBE given in Figure 5 exists i¤
L(c=vH ) + L(e) + L(a) · L(x) · L(c=vH ) + L(e) ¡ L(a); the Good
Pool equilibrium exists i¤ L(x) ¸ L(c=vH ) ¡ L(a), and the Bad Pool
equilibrium exists i¤ L(x) · L(c=vH ) + L(a):
proof. We’ve already shown that L(x) · L(c=vH ) + L(e) ¡ L(a)
motivates Other to play D when he sees s = 0; the demonstration is
similar that L(c=vH ) + L(e) + L(a) · L(x) motivates him to play C
Equilibrium Vengeance
14
when s = 1: This veri…es the DC part of the separating PBE in Figure
5. Given this behavior by Other, Self with v = vH will face D with
probability ® = (1 ¡ e)a + e(1 ¡ a) = a + e ¡ 2ae when playing T; a
simple calculation shows that the expected payo¤ is nonnegative (and
therefore she will try to play T) as long as ® · 1=(2 + v): Self with
v = 0 will face D with probability 1 ¡ ® when playing T; and she will
avoid doing so as long as ® · 1=(2 + v) = 1=2; a redundant condition.
This completes the separating PBE veri…cation.
If Other …nds it worthwhile to ignore the s = 0 signal because
unvengeful types are quite rare, then those types will also try to play
T. In the Bayes theorem calculation the expression (1 ¡ x)e(1 ¡ a)
thus is replaced by (1 ¡ x)(1 ¡ e)(1 ¡ a) and the factors involving e
cancel. Thus the condition L(x) ¸ L(c=vH ) ¡ L(a) ensures that Other
wants to play C even when s = 0: The condition ensuring that Self
indeed wants to play T is the same as before taking a = 0; so it holds
a fortiori. This completes the Good Pool PBE veri…cation.
Finally, if Other …nds it worthwhile to ignore the s = 1 signal
because vengeful types are quite rare, then those types will also try to
play N. In the Bayes theorem calculation the factors involving e again
drop out, and the condition L(x) · L(c=vH )+L(a) ensures that Other
prefers to play D even when s = 1: The condition e < 1=2 ensures that
both types of Self prefer to play N when Other always tries to play D.
This completes the Bad Pool PBE veri…cation and the proof.///
Equilibrium Vengeance
15
PBE
Parameter Values: a = 0.1, e = 0.05, c = 0.5, vH = 2
No Separating or
Pooling PBE
Separating
Good
Pooling
Bad
Pooling
x = 0.136
L = - 1.85
x=0
x = 0.002
L = - 6.24
x = 0.036
L = - 3.30
x = 0.75
L = 1.10
x=1
Separating Equilibrium
Vengeful type, v = vH
Non-vengeful type, v = 0
Fitness function
(1 – e)(1 – (2 + vH)a)
e(2a – 1)
Value in example
0.418
- 0.036
Good Pooling Equilibrium
Vengeful type, v = vH
Non-vengeful type, v = 0
Fitness function
(1 – e)(1 – (2 + vH)e)
(1 – e)(1 – 2e)
Value in example
0.760
0.855
Figure 6: PBE
A numerical example may help …x ideas. Figure 6 takes the marginal punishment cost to be 0:5 and the vengeful type to have preferred
cost vH = 2:0: It assumes behavioral noise rate e = 0:05 and observational noise rate a = 0:10: Then for su¢ciently small proportions of
the vengeful type (L(x) · ¡3:30 or x · 0:036) we have a Bad Pooling equilibrium in which both types of Self try to opt out and Other
tries to defect regardless of signal. For an overlapping range of venge-
Equilibrium Vengeance
16
ful type proportions (L(x) 2 (¡6:24; ¡1:85) or x 2 (0:002; 0:136)) we
have the Separating equilibrium. No separating or pooling PBE exists
for higher values of x until we reach x = 3=4 after which point we have
the Good Pooling equilibrium.
5.
EPBE
We do not yet have an evolutionary equilibrium because the types
v = 0 and vH and their population fractions x and 1¡x are exogenous.
Types that achieve higher …tness should displace the less …t; in evolutionary equilibrium all existing types within any population should
achieve equal and maximal …tness. In the numerical example, the
vengeful type in the Separating PBE has …tness (1 ¡e)(1¡(2+v)®) =
.95(1-4*.14) = 0:418, while the unvengeful type has …tness e(2®¡1) =
.05(.28-1) = -0.036. Thus we should expect the fraction x of the more
successful vengeful types to increase (assuming that x somehow is
positive to begin with, an issue to be discussed later). A su¢cient
increase in x will hit the upper bound of the separating PBE and
motivate Other to play C irrespective of the signal. However, in the
Good Pool equilibrium the unvengeful type is …tter; from Table 1 we
see that W S;GP (vH ) = (1 ¡ e)(1 ¡ (2 + vH )e) while the unvengeful
type achieves strictly greater …tness W S;GP (0) = (1 ¡ e)(1 ¡ 2e): In
the numerical example the …tnesses are .95*.8 = 0.760 for the vengeful
vs .95*.9 = 0.855 for the unvengeful. Thus we should expect that x
will decrease in the Good Pool PBE. In neither case do we have an
evolutionary equilibrium.
A few minutes re‡ection suggests that there may be a hybrid equilibrium in which some fraction q of the Other population attends the
signal (i.e., tries to play DC) while the rest don’t (i.e., (1 ¡ q) of Others try to play CC). This fraction q must be such that unvengeful and
vengeful Selfs achieve equal payo¤, and the fraction x of vengeful Selfs
must be such that Others achieve the same …tness whether or not they
attend to the signal. The intuition can be seen in Figure 7: by mixing
the …tness landscape in panel A with that in panel B, one can hope to
Equilibrium Vengeance
17
get the landscape in panel C with equal local (and global!) maxima
at v = 0 and vH :
F itn e s s W a s a F u n c tio n o f V e n g e fu ln e s s v
S e p a ratin g
E q u ilib r iu m
1
(q )
0
c
vH
G o o d P o o lin g
E q u ilib r iu m
1
(1 - q )
0
vH
H y b rid E P B E
1
(q ,1 - q )
0
vH
Figure 7: Fitness W as a Function of Vengefulness v
Working out such an equilibrium one encounters several complications.
² The unvengeful type Selfs must behave coherently. In the simple
mix of equilibria some try to play T expecting C and a positive
payo¤ while others try to play N expecting D and a negative
payo¤. This is incoherent because those unvengeful Selfs who
Equilibrium Vengeance
18
expect C are no more likely than those who expect D to match
up with the Others who don’t attend the signal.
² In EPBE all unvengeful Selfs must either all try to play T, or all
try to play N, or all must be indi¤erent. It can be shown that
indi¤erence is possible only if Self’s …tness is 0, and in EPBE
the vengeful type would also have …tness zero. Likewise, all
unvengeful Selfs try to play N only when their …tness is negative.
Since we are interested in mutual social gains, we seek an EPBE
with positive …tness and therefore where all unvengeful as well
as vengeful Selfs try to play T.
² The separating PBE calculation for the threshold x also needs
modi…cation, since it assumes di¤erent initial choices for vengeful
and nonvengeful Selfs.
² More fundamentally, the value of vH is no longer exogenous. It
should maximize the …tness of the vengeful type (again, in the
hybrid equilibrium, not necessarily in the separating PBE).
² The error amplitude of the signal should decrease in vH ; extremely vengeful types should be easier to distinguish from v = 0
types than slightly vengeful types. Thus we also need to take a
deeper look at the signal technology when vH is endogenous.
To sort this out, we begin with a general de…nition of evolutionary
equilibrium. It is written in English to avoid heavy notation that
won’t be needed later.
De…nition 1 An evolutionary perfect Bayesian equilibrium (EPBE)
is a PBE distribution over (extensive form game) strategy pro…les such
that in each population the strategies in the support of the distribution
achieve equal and maximal expected …tness.
The de…nition appears to be new. The novelty is that the distribution over types is endogenous. Most literature so far on evolution in
Equilibrium Vengeance
19
games of incomplete information allows di¤erent types to have di¤erent …tnesses and arbitrary distribution. (See Jacobsen et al, 2001, for a
recent example.) Although the justi…cation is seldom spelled out, the
idea seems to be that the types are determined by last minute circumstance, and evolutionary selection applies to complete type-contingent
strategies rather than to the types themselves. For some applications
this might be appropriate: when an animal …nds itself at di¤erent
points of its life to be the large or small type, or the owner or intruder
type, for example. For other cases, including the signalling game in
Jacobsen et al, 2001 it seems hard to escape the interpretation that
they are studying temporary or short run equilibria and in the longer
run the high types (who obtain higher equilibrium payo¤) will become
relatively more prevalent.
Returning now to the speci…c game shown in Figure 4, we need to
augment Self’s strategy set to include the entire range of vengeance
types, so Self maximizes …tness in fv 2 [0; v max ]g£fPr[T ] 2 [e; 1¡e]g:
Likewise, in EPBE Other uses a strategy that maximizes …tness in
fPr[Cjs = 0] 2 [e; 1 ¡ e]g £ fPr[Cjs = 1] 2 [e; 1 ¡ e]g. The other
modi…cation is that the observational error rate a now depends on the
distance from v = 0: We assume a smooth function a(v) that is positive
and decreasing, with a(0) = 1=2 (i.e., in the limit as the vengeful type
becomes completely unvengeful, the types can’t be distinguished) and
a(v) ! 0 as v ! 1:
There is a trivial EPBE related to Bad Pool PBE in which vengeful
types are so rare that Other always tries to play D and all types of
Self try to play N. In such PBE the less vengeful types are more
…t, and the vengeful types become extinct. Evolution then produces
a Self population that all has v = 0 and tries to play N, and the
Other population always tries to play D regardless of signal. It is
straightforward to check that this is indeed an EPBE.
We now seek a positive …tness hybrid EPBE where the Self population distribution has support consisting of two points f0; vH g with
mass (1 ¡ x) and x, and Pr[T ] = 1 ¡ e for both types; and the
Other population distribution has support consisting of the two points
Equilibrium Vengeance
20
(1 ¡ e; 1 ¡ e) and (e; 1 ¡ e) with mass (1 ¡ q) and q. Our intuition on
dynamics (mentioned above) suggests that such an equilibrium would
have a large basin of attraction including the uniform distributions
over the feasible strategy pro…les and all nearby distributions. For
such an EPBE we have the following necessary conditions.
vH = arg max
fEq W S (vja(v); e)g;
max
(1)
Eq W S (0ja(0); e) = Eq W S (vH ja(vH ); e), and
(2)
Ex W O ((1 ¡ e; 1 ¡ e)ja(v); e) = Ex W O ((e; 1 ¡ e)ja(v); e):
(3)
v2[c;v
]
The …rst condition says that the vengeful type achieves maximal
…tness given the exogenous tremble rate e, the endogenous signal error amplitude a(v), and the equilibrium attention probability q. The
restriction v 2 [c; v max ] re‡ects the ine¢cacy of positive v less than c,
and some biological limit v max >> c discussed in our earlier papers.
From Table 1 (temporarily called Figure 6) one sees that vengeful types
trying to play T attain …tness Eq W S (v) = (1 ¡ e)[q(1 ¡ ® ¡ (1 + v)®)
+ (1 ¡ q)((1 ¡ e) ¡ (1 + v)e)] = (1 ¡ e)[1 ¡ (2 + v)e ¡ qa(2 + v)(1 ¡ 2e)]:
The …rst order condition 0 = dEq W S =dv simpli…es slightly to 0 =
¡e ¡ qa0 (2 + v)(1 ¡ 2e) ¡ qa(1 ¡ 2e) or
[
e
]q ¡1 = ¡(2 + v)a0 ¡ a
1 ¡ 2e
(4)
We consider a simple Gaussian parametric form for the signal error
amplitude, with precision parameter k > 0:
a(v) = 0:5 exp(¡kv 2 ); so a0 = ¡2kva:
(5)
Substituting the a0 expressions in the …rst order condition (4) we obtain
[
1
e
]q ¡1 = a[2kv(2 + v) ¡ 1] = (kv(2 + v) ¡ ) exp(¡kv2 ):(6)
1 ¡ 2e
2
Equilibrium Vengeance
21
00
a
The maximization problem (1) has second order condition ¡a
0 ¸
2
0
00
, where ¡a > 0 and a ¸ 0: In the case of (5), the condition can
2+v
be written k ¸ 2v3v+2
2 (2+v) :
The second condition (2) says that vengeful and nonvengeful type
Selfs coexist in the EPBE because they have equal …tness. Recall
that Eq W S (v) = (1 ¡ e)[1 ¡ (2 + v)e ¡ qa(2 + v)(1 ¡ 2e)]: Recall
also that we are looking for an EPBE in which even the unvengeful
play T, so Eq W S (0) = (1 ¡ e)[q(® ¡ (1 ¡ ®)) + (1 ¡ q)((1 ¡ e) ¡ e)]
= (1 ¡ e)[1 ¡ 2e ¡ q(2 ¡ 2a ¡ 4e + 4ae)]: Thus (2) reduces to ve =
q[2(1 ¡ 2®) ¡ av(1 ¡ 2e)] = q(1 ¡ 2e)[2 ¡ a(4 + v)]: We obtain the
necessary condition for Other’s mixture
[
e
]q ¡1 = (2 ¡ a(4 + v))=v:
1 ¡ 2e
(7)
The third condition (7) says that Others who attend the signal
coexist in EPBE with those that don’t. Since all unvergeful types in
the candidate equilibrium try to play T, the error rate drops out for
Self. The relevant error rate for Other is ® = a + e ¡ 2ae: Calculations
similar to those in the Good Pool PBE lead to
v¡c
®
)(
)):
c
1¡®
(8)
e + qa(1 ¡ 2e) · 1=(2 + vH ):
(9)
x = 1=(1 + (
One algorithm for calculating EPBE (and verifying existence and
uniqueness) is to note that (4) and (7) have the same left hand sides,
so one can solve for v by equating their right hand sides. One then
obtains q from either equation, a from its de…nition, and x from (8). To
con…rm that vengeful Selfs (and a fortiori unvengeful Selfs) prefer T,
one checks that Eq W S (v) is positive, i.e., 1 ¸ (2+v)e+qa(2+v)(1¡2e)
or
Note that (9) is the same as the corresponding condition in PBE except
that ® = e + a(1 ¡ 2e) is replaced by an expression using qa instead
of a: Hence (9) is easier to satisfy than the PBE condition.
Equilibrium Vengeance
22
Equating the RHS of (6) and (7) we get 2kv(2 + v)a ¡ a = (2 ¡
4a)=v ¡ a or kv3 + 2kv 2 + 2 = 1=a = 2 exp(kv 2 ): At v = 0 both sides
of this last equation are equal to 2, and have equal slopes of 0. The
LHS has slope 4kv(1 + 43 v) and the RHS has slope 4kv exp(kv2 ) =
4kv(1 + kv 2 + :::):For for small positive v (up to approximately v =
3
) the LHS has steeper slope but the reverse is true for larger v
4k
(indeed, the slope ratio becomes in…nite), so the equation has a unique
solution for positive v. Substituting this solution vH into (6) one
obtains a positive value of q for each choice of positive e as long as
1
k isn’t too small (viz, we must have v(2 + v) > 2k
). The value of q
must also be less than 1.0 in EPBE, which is ensured by a tremble
1
rate e < 2+1=c
;where c is the positive value of the RHS of (6). One
completes the calculation and checks of the EPBE as described above.
For example, take c = 0:5, e = 0:05; and k = 0:28: Then v = 2:178;
a = 0:132; q = 0:1024 and x = 0:66: The second order condition holds
since 0.28>8.534/39.64=.215. The condition (9) ensuring that T is
preferred holds since .061<1/4.178=.24.
The range and behavior of the EPBE equilibrium can be explored
by varying the precision parameter k, while keeping c and e …xed,
as shown in Figure 8. Condition (9) and the other conditions are
easily seen to hold. In fact, for k slightly above 0.51, the second order
condition that accompanies (5) is violated for the given parameter
values. Also, it appears that k cannot approach 0 without violating
the constraint that q must be between 0 and 1. In each case, the
hybrid EPBE (i.e., excluding the trivial case where v is always 0 and
Self always chooses not to trust) is unique.
Equilibrium Vengeance
23
Table: EPBE (c=0.5, e=0.05)
k
0.1
0.2
0.28
0.5
0.51
v
4.502
2.782
2.178
1.402
1.380
a
0.066
0.106
0.132
0.187
0.189
q
0.347
0.121
0.102
0.078
0.078
x
0.505
0.562
0.594
0.665
0.668
Figure 8: EPBE (c=0.5, e=0.05)
The example suggests that, as the precision of Other’s signal, measured by k, increases, the equilibrium value of the vengeance ’parameter’ declines monotonically. This tends to increase the equilibrium
signal error rate, a, but not too rapidly. The equilibrium rate at which
Other attends to the signal, q, is quite small, and is also monotonically decreasing as the precision of the signal, k, increases in this example. Finally, the equilibrium fraction of vengeful Selfs, x, increases
monotonically in the precision parameter.
6.
Discussion
At least three important issues remain. First, is the interesting
EPBE stable? Preliminary work suggests that it is not unstable, but
has the same issues of neutral stability as do unique mixed Nash equilibria in simple 2x2 bimatrix games. More speci…cally, our current conjecture is that most reasonable evolutionary dynamics for our model
will, for a wide range of the basic parameters k (the signal precision
parameter), e (the tremble or behavioral noise amplitude), and c (the
marginal punishment cost) have two EPBE. Viz,
² The trivial (and ine¢cient) EPBE will have an open basin of
attraction and so will be the long-run fate of situations where
Equilibrium Vengeance
24
most Selfs initially are not very vengeful and most Others try to
play D, at least when they get the s = 0 signal.
² The hybrid EPBE (which reaps a major portion of the potential
mutual gains) will have a one dimensional stable manifold and a
two dimensional center manifold with a large basin of attraction.
A second issue springs from the trivial EPBE: how can one get
a critical mass to escape from the trivial EPBE? Or more simply, in
the context of the simple game in Figure 2, how can one get v > c
starting from v = 0? In another paper we suggest a possible answer
to this threshold problem. Subthreshold v < c is not adaptive in
a large population, but in small groups one can show that it works
together with the discount factor ± to increase …tness. Thus positive
values of v could get started in smaller groups and eventually become
advantageous in larger groups.
Third, one should consider endogenizing the other parameters.
Keeping punishment technology c constant (or doing comparative exercises) seems to make sense. The tremble rate parameter e trades o¤
trivially against the endogenous probability q that Other attends to
the signal, as can be seen from equations (7) and (4). However, there
is every reason to take seriously the evolution of the signal technology.
A mutant Self with true vengeance parameter v = 0 who could sometimes mimic the vengeful type would receive a major …tness boost. We
refer to this in another paper as the Viceroy problem, a reference to
Monarch butter‡ies who correspond to vengeful types and their mimics known as Viceroys. The other paper o¤ers an elaborate solution
to this problem involving interactions within and across small groups.
In the present paper we focused on two ideas, each of which we
believe has widespread applicability independent of the other. Emotional state dependent utility components (ESDUCs) o¤er a tractable
and ‡exible modelling approach to other-regarding preferences that
is capable of dealing with many of the leading issues in behavioral
economics. In particular, the vengeful components emphasised in the
present paper may be the key ingredient in models giving new insights
Equilibrium Vengeance
25
into ”irrational” con‡icts ranging from employment relations to international struggles. Friendly components may enter models providing
insight into behavior within the family, teams, charitable giving, etc.
Our emphasis has been a discipline on such other regarding components: they must directly or indirectly bring evolutionary …tness. This
theoretical discipline, together with the empirical discipline already
favored by behavioral economists, should help to sharpen behavioral
models.
The second idea is evolutionary perfect Bayesian equilibrium (EPBE).
We wrote a general verbal de…nition and worked it out explicitly for a
particular (and not especially simple) game of incomplete information.
We believe that EPBE is an appropriate characterization of long run
behavior when there are multiple ”types” and some opportunity for
entry, exit and/or switching among types. Many games of incomplete
information could be reconsidered in this light.
References
JJS01 Jacobsen HJ, Jensen M and Sloth B, ”Evolutionary Learning in
Signalling Games,” Games and Economic Behavior 34:1 (January
2001) pp34-63.