Frustration and Anger in Games

52 Pages·2015·0.5 MB·English

Frustration and Anger in Games
(cid:3)
Pierpaolo Battigalli Martin Dufwenberg Alec Smith
y z x
August 24, 2015
Abstract
Frustration, anger, and aggression have important consequences
for economic and social behavior, concerning for example monopoly
pricing, contracting, bargaining, tra¢ c safety, violence, and politics.
Drawingoninsightsfrompsychology, wedevelopaformalapproachto
exploring how frustration and anger, via blame and aggression, shape
interaction and outcomes in economic settings.
KEYWORDS: frustration, anger, blame, belief-dependent prefer-
ences, psychological games
JEL codes: C72, D03
This paper modi(cid:133)es and extends Smith (2009). Pierpaolo Battigalli gratefully ac-
(cid:3)
knowledges(cid:133)nancialsupportfromERCadvancedgrant324219. WethankPaoloLeonetti
andMarcoStenborgPettersonforexcellentresearchassistance,andDougBernheim,Steve
Brams, Vince Crawford, Nicodemo De Vito, Pierfrancesco Guarino, Michele Griessmair,
Botond K‰oszegi, Joshua Miller, David Rietzke, Julio Rotemberg, Emanuel Vespa, and
multiple seminar and conference audiences for helpful comments.
Bocconi University and IGIER. Email: [email protected].
y
University of Arizona, Bocconi University, IGIER, University of Gothenburg, CESifo.
z
Email: [email protected].
Virginia Tech, Department of Economics. Email: [email protected].
x
1
1 Introduction
Anger can shape economic outcomes in important ways. Consider three
cases:
Case 1: Petroleum costs sky-rocketed in 2006. Did gas stations
hold prices steady to avoid accusations of price gouging? Where
prices rose, did this cause road rage?
Case 2: When local football teams that are favored to win in-
stead lose, the police get more reports of husbands assaulting
wives(Card&Dahl2011). Dounexpectedlossesspurthusvented
frustration?
Case 3: FollowingSovereignDebtCrises(2009-), someEUcoun-
triesembarkedonausterityprograms. Wasitbecausecitizenslost
bene(cid:133)ts that some cities experienced riots?
Tra¢ c safety, pricing, domestic violence, political landscapes: the cases
illustrate situations where anger has important consequences. However, to
carefully assess how anger may shape social and economic interactions, one
needsatheorythatpredictsoutcomesbasedonthedecision-makingofanger-
prone individuals and that also accounts for the strategic consideration of
such individuals(cid:146)behavior by their co-players. We develop such a theory.
Insights from psychology about both the triggers of anger and its conse-
quences for behavior are evocative. The behavioral consequences of emotions
arecalled(cid:147)actiontendencies,(cid:148)andtheactiontendencyassociatedwithanger
is aggression. Angry players may be willing to forego material gains to pun-
ish others, or be predisposed to behave aggressively when this serves as a
credible threat, and so on. But while insights of this nature can be gleaned
from psychologists(cid:146)writings, their analysis usually stops with the individual
rather than going on to assess overall economic and social implications. We
take the basic insights about anger that psychology has produced as input
and inspiration for the theory we develop.1
1The relevant literature is huge. A source of insights and inspiration for us, is the
International Handbook of Anger (Potegal, Spielberger & Stemmler 2010), which o⁄ers a
cross-disciplinary perspective re(cid:135)ecting (cid:147)a⁄ective neuroscience, business administration,
epidemiology, health science, linguistics, political science, psychology, psychophysiology,
and sociology(cid:148)(p. 3). The non-occurrence of (cid:147)economics(cid:148)in the list may indicate that
our approach is original!
2
Anger is typically anchored in frustration, which occurs when someone
is unexpectedly denied something he or she cares about.2 We assume (ad-
mittedly restrictively; cf. Section 7) that people are frustrated when they
get less material rewards than they expected beforehand. Moreover, they
then become hostile towards whomever they blame. There are several ways
that blame may be assigned (cf. Alicke 2000) and we present three distinct
approaches, captured by distinct utility functions. While players motivated
by simple anger (SA) become generally hostile when frustrated, those mo-
tivated by anger from blaming behavior (ABB) or by anger from blaming
intentions (ABI) go after others more discriminately, asking who caused, or
who intended to cause, their dismay.
To provide general predictions, we develop a notion of polymorphic se-
quential equilibrium (PSE). Players correctly anticipate how others behave
on average, yet di⁄erent types of the same player may have di⁄erent plans
in equilibrium. This yields meaningful updating of players(cid:146)views of others(cid:146)
intentions as various subgames are reached, which is crucial for a sensible
treatment of how players consider intentionality as they blame others. We
apply this solution concept to the aforementioned utility functions, explore
properties, and compare predictions.
A player(cid:146)s frustration depends on his beliefs about others(cid:146)choices. The
blame a player attributes to another may depend on his beliefs about oth-
ers(cid:146)choices or beliefs. For these reasons, all our models (cid:133)nd their intellec-
tual home in the framework of psychological game theory; see Geanakoplos,
Pearce & Stacchetti (1989), Battigalli & Dufwenberg (2009).
Several recent studies inspire us. Most are empirical, indicative of hostile
action occurring in economic situations, based on either observational or
experimental data.3 A few studies present theory, mostly with the purpose
of explaining speci(cid:133)c data patterns (Rotemberg 2005, 2008, 2011; Akerlof
2013; Passarelli & Tabellini 2013). Our approach di⁄ers in that we do not
start with data, but with notions from psychology which we incorporate into
general games, and we are led to use assumptions which di⁄er substantially
(Section 7 elaborates, in regards to Rotemberg(cid:146)s work). Brams (2011), in his
bookGameTheoryandtheHumanities,buildingonhisearlier(1994)(cid:147)theory
of moves,(cid:148)includes negative emotions like anger in the analysis of sequential
2Psychologists often refer to this as (cid:147)goal-blockage;(cid:148)cf. p.3 of the (op.cit.) Handbook.
3SeeAnderson&Simester(2010)andRotemberg(2005,2011)onpricing,Card&Dahl
on domestic violence, and Carpenter & Matthews (2012), Gurdal, Miller & Rustichini
(2014), and Gneezy & Imas (2014) for experiments.
3
interaction. Speci(cid:133)cally, he considers players who alternate in changing the
state of a two-by-two payo⁄ matrix, with payo⁄s materializing only at the
terminal state. His analysis di⁄ers from ours because it is restricted to this
class of games and he does not assume that anger is belief-dependent.
We develop most of our analysis for a two-period setting described in
Section 2. Section 3 de(cid:133)nes frustration. Section 4 develops our notions of
psychological utility. Section 5 introduces and explores equilibriumbehavior.
Section 6 generalizes the analysis to multistage games. Section 7 concludes.
Proofs of results are collected in an online appendix.
2 Setup
We begin by describing the rules of interaction (the game form), and then
we de(cid:133)ne beliefs.
2.1 Game form
We consider a (cid:133)nite two-stage game form describing the rules of interaction
and the consequences of players(cid:146)actions. The set of players is I. To ease
notation, we assume that all players take actions simultaneously at each
stage. Thus, nodes are histories of action pro(cid:133)les at = (at) ; h = ? is
i i I
the empty history (the root of the game), h = (a1) is a hist2ory of length
one, which may be terminal or not, and h = (a1;a2) is a history of length
2, which is terminal. H is the set of non-terminal histories and Z is the
set of terminal histories. The set of feasible actions of i given h H is
2
A (h). This set is a singleton if i is not active given h. Thus, for h H,
i
2
I(h) = i I : A (h) > 1 is the set of active players given h. In a perfect
i
f 2 j j g
information game I(h) is a singleton for each h H. We omit parentheses
2
whenever no confusion may arise. For example, we may write h = a1 instead
of h = (a1), and h = (a1;a2) if i (resp. j) is the only (cid:133)rst (resp. second)
i j
mover. Finally, we let A(h) = A (h) and A (h) = A (h).
i I i i j=i j
(cid:2)2 (cid:0) (cid:2) 6
We assume that the material consequences of players(cid:146)actions are deter-
mined by a pro(cid:133)le of monetary payo⁄functions ((cid:25)i : Z ! R)i I. This com-
pletes the description of the game form, if there are no chance2moves. If the
gamecontainschancemoves, weaugmenttheplayersetwithadummyplayer
c (with c = I), who selects a feasible action at random. Thus, we consider
2
an augmented player set I = I c , and the sets of (cid:133)rst and second movers
c
[f g
4
may include c: I(?);I(a1) Ic. If the chance player is active at h H, its
(cid:18) 2
move is described by a probability density function (cid:27) ( h) (cid:1)(A (h)).
c c
(cid:1)j 2
Thefollowingexample, towhichwewillreturninourdiscussionofblame,
is here employed to illustrate our notation:
Figure A. Asymmetric punishment.
Example 1 Ann and Bob (a and b in Figure A) move simultaneously in
the (cid:133)rst stage. Penny the punisher (p in Figure A) may move in the second
stage; by choosing P she then decreases (cid:25) (while (cid:25) increases). See Figure
b a
A. Pro(cid:133)les of actions and monetary payo⁄s are listed according to players(cid:146)
alphabetical order. We have:
H = ?;(D;L) , Z = (U;L);(U;R);(D;R);((D;L);N);((D;L);P) ,
f g f g
I(?) = a;b , I((D;L)) = p ,
f g f g
Aa(?) = U;D , Ab(?) = L;R , Ap((D;L)) = N;P . N
f g f g f g
2.2 Beliefs
It is conceptually useful to distinguish three aspects of a player(cid:146)s beliefs:
beliefs about co-players(cid:146)actions, beliefs about co-players(cid:146)beliefs, and the
player(cid:146)s plan which we represent as beliefs about own actions. Beliefs are
5
de(cid:133)ned conditional on each history. Abstractly denote by (cid:1) the space of
i
(cid:0)
co-players(cid:146)beliefs (the formal de(cid:133)nition is given below). Player i(cid:146)s beliefs
can be compactly described as conditional probability measures over paths
and beliefs of others, i.e., over Z (cid:1) . Events, from i(cid:146)s point of view, are
i
(cid:2) (cid:0)
subsets of Z (cid:1) . Events about behavior take form Y (cid:1) , with Y Z;
i i
events about(cid:2)beli(cid:0)efs take form Z E , with E (cid:1)(cid:2) .4(cid:0) (cid:18)
(cid:2) (cid:1)(cid:0)i (cid:1)(cid:0)i (cid:18) (cid:0)i
Personal histories To model how i determines the subjective value of
feasible actions, we add to the commonly observed histories h H also
2
personal histories of the form (h;a ), with a A (h). In a game with perfect
i i i
2
information, (h;a ) H Z. But if there are simultaneous moves at h,
i
2 [
then (h;a ) is not a history in the standard sense. As soon as i irreversibly
i
chooses action a , he observes (h;a ), and can determine the value of a using
i i i
his beliefs conditional on this event (i knows in advance how he is going to
update his beliefs conditional on what he observes). We denote by H the set
i
of histories of i (cid:151)standard and personal(cid:151) and by Z(h ) the set of terminal
i
successors of h .5 The standard precedence relation for histories in H Z
i
(cid:30) [
is extended to H in the obvious way: for all h H, i I(h), and a A (h),
i i i
2 2 2
it holds that h (h;a ) and (h;a ) (h;(a ;a )) if i is not the only active
i i i i
(cid:30) (cid:30) (cid:0)
player at h. Note that h h implies Z(h) Z(h), with strict inclusion if
0 0
(cid:30) (cid:18)
at least one player (possibly chance) is active at h.
First-order beliefs For each h H , player i holds beliefs (cid:11) ( Z(h ))
i i i i
2 (cid:1)j 2
(cid:1)(Z(h )) about the actions that will be taken in the continuation of the
i
game. The system of beliefs (cid:11) = ((cid:11) ( Z(h ))) must satisfy two prop-
i i (cid:1)j i hi Hi
erties. First, the rules of conditional probabilities2hold whenever possible: if
h h then for every Y Z(h )
i (cid:30) 0i (cid:18) 0i
(cid:11) (Y Z(h ))
i i
(cid:11) (Z(h ) Z(h )) > 0 (cid:11) (Y Z(h )) = j . (1)
i 0i j i ) i j 0i (cid:11) (Z(h ) Z(h ))
i 0i j i
4(cid:1) turns out to be a compact metric space. Events are Borel measurable subsets of
i
Z (cid:1)(cid:0) . We do not specify terminal beliefs of i about others(cid:146)beliefs, as they are not
i
rel(cid:2)evan(cid:0)t for the models in this paper.
5That is, H = H (h;a ):h H;i I(h);a A (h) . The de(cid:133)nition of Z(h ) is
i i i i i
[f 2 2 2 g
standard for h H; for h =(h;a ) we have Z(h;a )= Z(h;(a ;a )).
i i i i i i
2 a(cid:0)i2SA(cid:0)i(h) (cid:0)
6
We use obvious abbreviations to denote conditioning events and the condi-
tional probabilities of actions: for all h H, a = (a ;a ) A (h) A (h),
i i i i
2 (cid:0) 2 (cid:2) (cid:0)
(cid:11) (a h) = (cid:11) (Z(h;a) Z(h)),
i i
j j
(cid:11) (a h) = (cid:11) (a ;a h),
i;i ij i i 0 ij
(cid:0)
a0(cid:0)i2XA(cid:0)i(h)
(cid:11) (a h) = (cid:11) (a ;a h).
i;(cid:0)i (cid:0)ij i 0i (cid:0)ij
a0i2XAi(h)
Note that (cid:11)i(ai h) = (cid:11)i(Z(h;ai) Z(h)), and that (1) implies (cid:11)i(a1;a2 ?) =
j j j
(cid:11)i(a2 a1)(cid:11)i(a1 ?).
j j
With this, we can write in a simple way our second requirement, that i(cid:146)s
beliefs about the actions simultaneously taken by the co-players are indepen-
dent of i(cid:146)s action: for all h H, i I, a A (h), and a A (h),
i i i i
2 2 2 (cid:0) 2 (cid:0)
(cid:11) (a h) = (cid:11) (a h;a ). (2)
i; i i i; i i i
(cid:0) (cid:0) j (cid:0) (cid:0) j
Properties (1)-(2) imply
(cid:11) (a ;a h) = (cid:11) (a h)(cid:11) (a h).
i i i i;i i i; i i
(cid:0) j j (cid:0) (cid:0) j
Thus, (cid:11) is made of two parts, what i believes about his own behavior and
i
what he believes about the behavior of others. The array of probability mea-
sures (cid:11) (cid:1)(A (h)) is (cid:151)technically speaking(cid:151) a behavioral strategy,
i;i h H i
2 (cid:2) 2
and we interpret it as the plan of i. The reason is that the result of i(cid:146)s
contingent planning is precisely a system of conditional beliefs about what
action he would take at each history. If there is only one co-player, also
(cid:11) (cid:1)(A (h)) corresponds to a behavioral strategy. With multiple
i; i h H i
(cid:0) 2 (cid:2) 2 (cid:0)
co-players, (cid:11) corresponds instead to a (cid:147)correlated behavioral strategy.(cid:148)
i; i
(cid:0)
Whatever the case, (cid:11) gives i(cid:146)s conditional beliefs about others(cid:146)behavior,
i; i
(cid:0)
and these beliefs may not coincide with the plans of others. We emphasize:
a player(cid:146)s plan does not describe actual choices, actions on the path of play
are the only actual choices.
A system of conditional probability measures (cid:11) = ((cid:11) ( Z(h ))) that
i i (cid:1)j i hi Hi
satis(cid:133)es (1)-(2) is a (cid:133)rst-order belief of i. We let (cid:1)1 denote the s2pace of
i
such (cid:133)rst-order beliefs. It can be checked that (cid:1)1 is a compact metric space.
i
Hence, the same holds for (cid:1)1 = (cid:1)1, the space of (cid:133)rst-order beliefs
pro(cid:133)les of the co-players. (cid:0)i (cid:2)j6=i j
7
Second-order beliefs Players do not only hold beliefs about paths, they
also hold beliefs about the beliefs of co-players. In the following analysis, the
only co-players(cid:146)beliefs a⁄ecting the values of actions are their (cid:133)rst-order be-
liefs. Therefore, welimitourattentiontosecond-orderbeliefs, i.e., systems
of conditional probability measures ((cid:12) ( h )) (cid:1) Z(h ) (cid:1)1
that satisfy properties analogous to (1)i-((cid:1)j2)i.6 hFi2irHsit,2if(cid:2)hhi2Hih then i (cid:2) (cid:0)i
i (cid:30) 0i (cid:0) (cid:1)
(cid:12) (E h )
(cid:12) (h h ) > 0 (cid:12) (E h ) = i j i (3)
i 0ij i ) i j 0i (cid:12) (h h )
i 0ij i
for all h ;h H and every event E Z(h ) (cid:1)1 . Second, i realizes
i 0i 2 i (cid:18) 0i (cid:2) i
that his choice cannot in(cid:135)uence the (cid:133)rst-order beliefs o(cid:0)f co-players and their
simultaneous choices, so i(cid:146)s beliefs satisfy an independence property:
(cid:12) (Z(h;(a ;a )) E (h;a )) = (cid:12) (Z(h;(a ;a )) E (h;a )), (4)
i i (cid:0)i (cid:2) (cid:1)j i i 0i (cid:0)i (cid:2) (cid:1)j 0i
for every h H, a ;a A (h), a A (h), and event E (cid:1)1 about co-
players(cid:146)(cid:133)rs2t-orderibe0ili2efs.iThe s(cid:0)pia2ce o(cid:0)fisecond-order beli(cid:1)ef(cid:18)s of (cid:0)iiis denoted
(cid:1)2 .
i
(cid:0)It can be checked that starting from (cid:12) (cid:1)2 and letting (cid:11) (Y h ) =
i 2 i i j i
(cid:12) Y (cid:1)1 h for all h H and Y Z, we obtain a system (cid:11) satisfying
i (cid:2) ij i i 2 i (cid:18) i
(1)-(2), i.e(cid:0)., an element of (cid:1)1. This (cid:11) is the (cid:133)rst-order belief implicit in (cid:12) .
(cid:0) (cid:1) i i i
Whenever we write in a formula beliefs of di⁄erent orders for a player, we
assume that (cid:133)rst-order beliefs are derived from second-order beliefs, other-
wise beliefs of di⁄erent orders would not be mutually consistent. Also, we
write initial beliefs omitting the empty history, as in (cid:12) (E) = (cid:12) (E ?) or
i i
j
(cid:11)i(a) = (cid:11)i(a ?), whenever this causes no confusion.
j
Conditional expectations Let beanyreal-valuedmeasurablefunction
i
of variables that player i does not know, e.g., the terminal history or the co-
players(cid:146)(cid:133)rst-order beliefs. Then i can compute the expected value of
i
conditional on any common or personal history h H by means of his
i i
2
belief system (cid:12)i. This expected value is denoted E[ i hi;(cid:12)i]. If i depends
j
only on actions, i.e., on the path z, then E[ i hi;(cid:12)i] is determined by the
j
(cid:133)rst-order belief system (cid:11)i derived from (cid:12)i, and we can write E[ i hi;(cid:11)i]. In
j
particular, (cid:11) gives the conditional expected material payo⁄s:
i
6We use obvious abbreviations, such as writing h for event Z(h) (cid:1)1 , whenever this
causes no confusion. (cid:2) (cid:0)i
8
E[(cid:25)i h;(cid:11)i] = (cid:11)i(z h)(cid:25)i(z),
j j
z Z(h)
2X
E[(cid:25)i (h;ai);(cid:11)i] = (cid:11)i(z h;ai)(cid:25)i(z)
j j
z2ZX(h;ai)
for all h H, ai Ai(h). E[(cid:25)i h;(cid:11)i] is what i expects to get conditional on
2 2 j
h given (cid:11)i, which also speci(cid:133)es i(cid:146)s plan. E[(cid:25)i (h;ai);(cid:11)i] is i(cid:146)s expected payo⁄
j
of action a . If a is what i planned to choose at h, (cid:11) (a h) = 1, and then
i i i;i i
j
E[(cid:25)i h;(cid:11)i] = E[(cid:25)i (h;ai);(cid:11)i]. For initial beliefs, we omit h = ? from such
j j
expressions; in particular, the initially expected payo⁄is E[(cid:25)i;(cid:11)i].
3 Frustration
Anger is triggered by frustration. While we focus upon anger as a social
phenomenon (cid:151)frustrated players blame and become angry with and care for
the payo⁄s of others(cid:151) our account of frustration refers to own payo⁄s only.
In Section 7 (in hindsight of de(cid:133)nitions to come) we discuss this approach in
depth. Here, we de(cid:133)ne player i(cid:146)s frustration, in stage 2, given a1, as
+
Fi(a1;(cid:11)i) = (cid:20)E[(cid:25)i;(cid:11)i](cid:0)a2im2Aai(xa1)E[(cid:25)ij(a1;a2i);(cid:11)i](cid:21) ,
where[x]+ = max x;0 . Inwords, frustrationisgivenbythegap, ifpositive,
f g
betweeni(cid:146)sinitiallyexpectedpayo⁄andthecurrentlybestexpectedpayo⁄he
believes he can obtain. Diminished expectation (cid:151)E[(cid:25)i a1;(cid:11)i] < E[(cid:25)i;(cid:11)i](cid:151)
j
is only a necessary condition for frustration. For i to be frustrated it must
also be the case that i cannot close the gap.
F (a1;(cid:11) ) expresses stage-2 frustration. One could de(cid:133)ne frustration at
i i
the root, or at end nodes, but neither would matter for our purposes. At the
root nothing has happened, sofrustrationequals zero. Frustrationis possible
at the end nodes, but can(cid:146)t in(cid:135)uence subsequent choices as the game is over.
One might allow the anticipated frustration at end nodes to in(cid:135)uence earlier
decisions; however, the assumptions we make in the analysis below rule this
out. Furthermore, players are in(cid:135)uenced by the frustrations of co-players
only insofar as their behavior is a⁄ected.
9
Example 2 To illustrate, return to Figure A. Suppose Penny initially ex-
pects $2: (cid:11)p((U;L) ?) + (cid:11)p((D;R) ?) = 1 and E[(cid:25)p;(cid:11)p] = 2. After
j j
a1 = (D;L) we have
Fp((D;L);(cid:11)p) = [E[(cid:25)p;(cid:11)p] max (cid:25)p((D;L);N);(cid:25)p((D;L);P) ]+ = 2 1 = 1.
(cid:0) f g (cid:0)
This is independent of her plan, because she is initially certain she will not
move. If instead (cid:11)p((U;L)j?) = (cid:11)p((D;L)j?) = 12 then
1 1 1
F ((D;L);(cid:11) ) = 2+ (cid:11) (N (D;L)) 1 1 = (cid:11) (N (D;L));
p p p p
2 (cid:1) 2 j (cid:1) (cid:0) 2 j
Penny(cid:146)s frustration is highest if she initially plans not to punish Bob.
N
4 Anger
A player(cid:146)s preferences over actions at a given node (cid:151)his action tendencies(cid:151)
depend on expected material payo⁄s and frustration. A frustrated player
tends to hurt others, if this is not too costly (cf. Dollard et al. 1939, Averill
1983, Berkowitz 1989). We consider di⁄erent versions of this frustration-
aggression hypothesis related to di⁄erent cognitive appraisals of blame. In
general, player i moving at history h chooses action a to maximize the
i
expected value of a belief-dependent (cid:147)decision utility(cid:148)of the form
ui(h;ai;(cid:12)i) = E[(cid:25)i (h;ai);(cid:11)i] (cid:18)i Bij(h;(cid:12)i)E[(cid:25)j (h;ai);(cid:11)i], (5)
j (cid:0) j
j=i
X6
where (cid:11) is the (cid:133)rst-order belief system derived from second-order belief (cid:12) ,
i i
and (cid:18) 0 is a sensitivity parameter. B (h;(cid:12) ) 0 measures how much of
i ij i
(cid:21) (cid:21)
i(cid:146)s frustration is blamed on co-player j, and the presence of E[(cid:25)j (h;ai);(cid:11)i]
j
in the formula translates this into a tendency to hurt j. We assume that
B (h;(cid:12) ) is positive only if frustration is positive:
ij i
B (h;(cid:12) ) F (h;(cid:11) ). (6)
ij i i i
(cid:20)
Therefore, the decision utility of a (cid:133)rst-mover coincides with expected ma-
terial payo⁄, because there cannot be any frustration in the (cid:133)rst stage:
ui(?;ai;(cid:12)i) = E[(cid:25)ijai;(cid:11)i]. When i is the only active player at h = a1,
10