Frustration and Anger in Games
Pierpaolo Battigalli, Martin Dufwenberg and Alec Smith
Frustration & Anger in Games
(Extended abstract, prepared July 14, 2014)
Pierpaolo Battigalli*, Martin Dufwenberg** & Alec Smith***
* Bocconi University, IGIER
** Bocconi University, IGIER, University of Gothenburg, University of Arizona, CESifo
*** Caltech, Compass Lexecon
Introduction
It is not hard to think of examples where it seems plausible that anger may play a key role for
shaping outcomes in economically important ways. Consider some examples/questions:
1. In 2006 gas prices went up & up. Many folks were upset. Did this cause road rage?
2. When local football teams favored to win lose, the police get more reports of husbands
assaulting wives (Card & Dahl, 2011 QJE). Do unexpected losses spur thus vented frustration?
3. Following Sovereign Debt Crises (2009-), some EU countries embarked on austerity
programs. Was it because citizens lost benefits that many cities experienced riots?
Traffic safety, domestic violence, political landscapes, .... the themes seem important. However,
in order to systematically assess relevance and consequences one needs theory connecting anger,
decisions, and outcomes. In this paper we develop this.
Insights from psychology suggest ways that anger has strategic implications. The
behavioral consequences of emotions are referred to as “action tendencies,” and that of anger is
aggression. i One may imagine that angry players are willing to forego material gains to punish
others, or that a predisposition to behave aggressively when angered may benefit a player by
serving as a credible threat, and so on. But while insights of this nature can be gleaned from
psychologists’ writings, their analysis usually stops with the individual rather than going on to
assess overall economic implications. We take the basic insights about anger that psychology has
produced as input and inspiration for the theory we develop and apply. ii
Economists traditionally paid scant attention to anger, but interest is on the rise and
several recent studies greatly inspired us. Most are empirical, indicative of hostile action
occurring in economic situations, based on either archival iii or experimental iv data. A few present
theory, notably Rotemberg (2005 JME, 2008 JEBO, 2011 JEEA), Akerlof (2013, unpubl.),
Passerelli & Tabellini (2013, unpubl.), typically with the purpose of explaining specific data
patterns. Our approach is different. We do not start with data but with notions of psychology that
we incorporate in general games. We are lead to models that differ substantially from the
existing theory, though predictions may be similar in their specific settings.
More details
Psychologists suggest that anger is typically anchored in frustration, which occurs when
someone is unexpectedly denied something they care about. (Psychologists often refer to this as
"goal-blockage;" cf. p.3 of the op.cit. Handbook) We assume that people are frustrated when
they get less material rewards than they expected beforehand. How do decision-makers react to
frustration and anger? We assume that they become hostile towards whomever they blame.
However, there are several ways that blame may be assigned (cf. Alicke, 2000, Psych. Bull.) and
we present three distinct approaches, captured by distinct utility functions. While players
motivated by simple anger (SA) become generally hostile, those motivated 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.
What are the overall implications when people interact? To provide answers, we develop
a notion of polymorphic sequential equilibrium (PSE). Players are assumed to correctly
anticipate how others behave on average, and the concept furthermore allows for different
“types” of the same player to have different plans in equilibrium, which yields meaningful
updating of players’ views of others’ intentions as various subgames are reached. This is crucial
for a sensible treatment of how players consider intentionality as they blame others. We apply
this solution concept to the aforementioned utilities, explore properties, and compare predictions.
A player’s frustration depends on his beliefs about others’ choices. The blame a player
attributes to another may depend on his beliefs about others’ choices or beliefs. For these
reasons, all our models find their intellectual home in the framework of so-called psychological
game theory; see Geanakoplos, Pearce & Stacchetti (1989, GEB) and Battigalli & Dufwenberg
(2009, JET).
An example
It is impossible in an extended abstract to give detailed definitions and to conduct much analysis.
However, a glimpse of the nature of our models can be sensed by considering the following
example, involving a so-called “mini-ultimatum game” (mini-UG):
Player 1
/\
/ \
Fair/ \Greedy
/
\
/
\ Player 2
(2, 2)
/\
/ \
Reject/
\Accept
/
\
/
\
(0, 0)
(3, 1)
The given payoffs are material rewards, not utilities since preferences are affected by bursts of
anger that may set in. Several indications of properties of our solutions follow:
- Even if player 1 is, in principle, easily angered, in this particular game he will act as-if selfish,
and maximize expected monetary return. The reason is that since he cannot be negatively
surprised, moving only at the root, he cannot be frustrated. Experimenters have at times
documented behavior that seems to indicate that second-movers deviate more from selfish play
than first-movers; our model generates such a pattern endogenously.
- Utilities SA or ABB allow (Fair, Accept) as well as (Greedy, Reject) as PSE. On-path
rejections are compatible with a “mixed,” but monomorphic equilibrium where players plan to
make their choices with non-degenerate probabilities, and there is only one “type” of each
player. In such an equilibrium, for a given anger-sensitivity, player 1 is less likely to make the
greedy offer using the ABI notion then using SA or ABB. The reason is that under ABI player 2
will blame 1 less when he gets a Greedy offer, giving 1 some credit for having planned to choose
the Fair offer with a non-zero probability. By contrast, under SA or ABB 2 will blame 1 more,
and so be more prone to hostile response. Since the players must keep each other indifferent in a
mixed equilibrium there must be less frustration for 2, so 2 must be resigned to getting Greedy
offers with higher likelihood. Interestingly, the mixed monomorphic equilibrium under ABB is
equivalent to a partially mixed PSE under ABI with two types of player 1, the type who plans to
be Fair, and the type who plans to be Greedy. At the beginning player 2 is uncertain about 1’s
type, which determines his initially expected gain and later frustration, but after the Greedy offer
2 blames 1 for the intention to make such offer, and he is as angry toward 1 as under ABB.
- The more easily angered a player is (a personal trait), in a mixed equilibrium, the lower will be
the probabilities of that player getting a Fair offer. Again, to keep such a player indifferent his
frustration must be low, which will be the case if he is resigned to getting Greedy offers with a
high probability.
- [Reciprocity] Angry-response bears some similarity with negative reciprocity, but there are
crucial differences and the mini-UG illustrates one: (Greedy, Reject) cannot be a PSE for SA,
ABB, or ABI, but may be an (sometimes so-called “miserable”) equilibrium under reciprocity
theory.
- [External offers] If we consider the modification of the mini-UG where player 1 is replaced by
chance, or a disinterested third party (as did Blount, 1995 OBHDP), then under SA rejections
remain possible whereas they cannot occur under ABB or ABI utilities.
- [Cooling off] If we add a “waiting stage,” a post-Greedy-offer stage where player 2 has a
singleton choice set interpreted as her having to wait for a while before responding, then
prediction will depend on modeling details concerning frustrations. Does 2’s frustration depend
on her expected payoff at the root or at the previous stage? Since anger is transitory, depending
on how quickly a game is played, either assumption may make sense, and we develop versions of
our solutions for the variants. Solution change accordingly.
i
Baumeister & Bushman (2007, book, p. 66) e.g. say that “anger is an important and powerful cause of aggression”
defined (p. 62) as “any behavior that is intended to harm another person who is motivated to avoid the harm.”
ii
The relevant literature is huge. A good point of entry, and source of insights & inspiration for us, is the recent
International Handbook of Anger (Potegal, Stemmler & Spielberger, eds., 2010 Springer), which offers a crossdisciplinary perspective over 32 chapters reflecting “affective neuroscience, business administration, epidemiology,
health science, linguistics, political science, psychology, psychophysiology, and sociology” (p. 3, opening chapter).
We take the non-occurrence of “economics” in the list as an indication our approach is original and needed!
iii
iv
Anderson & Simester (2010, QJE) and Rotemberg (2005, 2011) on pricing; Card & Dahl on domestic violence.
Carpenter & Matthews (2012, JEEA), Fudenberg, Rand & Dreber (2012, AER), Gurdal, Miller & Rustichini
(2014, JPE), Gneezy & Imas (2014, PNAS).