1. society

Mälardalen
Mälardalen University
University Press
Press Dissertations
Dissertations
No.
156
No. 156
MATHEMATICAL MODELS OF SOCIAL
NORMS AND PETTY CORRUPTION
Alexander Funcke
2015
School
School of
of Education,
Education, Culture
Culture and
and Communication
Communication
Copyright © Alexander Funcke, 2015
ISBN 978-91-7485-187-8
ISSN 1651-4238
Printed by Arkitektkopia, Västerås, Sweden
Mälardalen University Press Dissertations
No. 156
Mälardalen University Press Dissertations
No. 156
MATHEMATICAL MODELS OF SOCIAL NORMS AND PETTY CORRUPTION
Alexander Funcke
MATHEMATICAL MODELS OF SOCIAL NORMS AND PETTY CORRUPTION
Alexander Funcke
Akademisk avhandling
som för avläggande av filosofie doktorsexamen i matematik/tillämpad matematik
vid Akademin för utbildning, kultur och kommunikation kommer att offentligen
försvaras fredagen den 27 mars 2015, 13.00 i Delta, Mälardalens högskola, Västerås.
Akademisk avhandling
Fakultetsopponent: professor David Sumpter, Uppsala Universitet
som för avläggande av filosofie doktorsexamen i matematik/tillämpad matematik
vid Akademin för utbildning, kultur och kommunikation kommer att offentligen
försvaras fredagen den 27 mars 2015, 13.00 i Delta, Mälardalens högskola, Västerås.
Fakultetsopponent: professor David Sumpter, Uppsala Universitet
Akademin för utbildning, kultur och kommunikation
Akademin för utbildning, kultur och kommunikation
Abstract
Corruption is a problem all around the world, but the extent of the problem varies between countries
and situations. In this thesis, I focus on how corruption levels can change when they are culturally
determined. For this reason, I study the dynamics of the cultural underpinnings: social norms and
conventions.
The dissertation consists of six papers. In the first paper, I expand a common definition of social norms.
The aim of the extension is to capture the fact that the scope of a social norm may be larger than just a
single specific situation. I introduce a similarity measure and develop a mathematical model according
to which all situations' social norms are interconnected, and affect each other, but those situations that
are most similar and most recent have the greatest normative effect on a current situation. Given this
model I test the effect of bringing about norm change by temporarily dismantling institutions and then
reestablishing them.
In the second paper, I show in a mathematical model how it is possible to design fine and reward
mechanisms that make it superfluous for individuals to form beliefs about how others will act. Through
this mechanism, it should be possible to circumvent the problem that norm change typically will be
successful only if it is synchronized across a large part of the population.
In the third paper, I and my co-authors, first conducted a survey. The results of which demonstrate
that there is a general tendency among people to consider themselves to be less prone to corrupt
behavior than the average person. Such an "everyone-is-better-than-average" effect is a well-established
phenomenon in social psychology but not previously demonstrated in the corruption domain. We then
show in a mathematical model that such systematic biases in estimation of own versus others' corruption
make it more difficult to achieve norm change in the direction of less corruption.
In the fourth and fifth paper we again consider the "everyone-is-better-than-average" effect and see how
in certain value based groups the effect can be reversed. This changes the insight from the third paper
slightly.
The last paper considers a classic question of how a collective can succeed in collective action when it
is risky to be among the first individuals to act. I and my co-author investigate how the collective can
benefit from access to a set of signal acts that signal an individual's level of commitment to the collective
cause. The problem is modeled as a threshold model where an individual's inclination to conduct a
specific act depends on the previous commitment level in the population.
ISBN 978-91-7485-187-8
ISSN 1651-4238
PA P E R S I N C L U D E D I N T H E S I S
1. Instilling norms in a turmoil of spillovers
Alexander Funcke
Published in PPE Working Papers, Philosophy, Politics and Economics,
University of Pennsylvania, 4 (2015)
2. A mechanism for optimal enforcement of coordination: sidestepping theory of mind
Alexander Funcke, Daniel Cownden
Published in PPE Working Papers, Philosophy, Politics and Economics,
University of Pennsylvania, 3 (2015)
3. Biased perception may trump rational intention: most people
think they are less corrupt than average
Alexander Funcke, Kimmo Eriksson, Pontus Strimling
Submitted to Rationality & Society
4. Humble self-enhancement: religiosity and the better-than-average
effect
Kimmo Eriksson, Alexander Funcke
Published in Social Psychological and Personality Science 5(1) (2014): 7683
5. A stairway to revolutionary collective below-average effect with
respect to american political stereotypes on warmth and competence
Kimmo Eriksson, Alexander Funcke
In press with Political Psychology. Published online before print September 30, 2013. doi: 10.1111/pops.12093
6. A stairway to revolutionary collective action
Alexander Funcke, Ulrik Franke
Submitted to Rationality & Society
3
ACKNO WLEDGMEN T
I am grateful to the Swedish Research Council, the Ratio Institute, the Edmond J. Safra Center for Ethics at Harvard University, the Behavioral Ethics
Lab at University of Pennsylvania, the Center for Study of Public Choice at
George Mason University and in particular the Centre for the Study of Cultural Evolution at Stockholm University, for hosting me, funding me and (at
least by proxy) having some belief in me.
I acknowledge Kimmo Eriksson and Pontus Strimling; my Ph.D. advisors
and co-authors. I tip my hat to my other co-authors Ulrik Franke and Daniel
Cownden. And I will keep tipping it to each and everyone who has worked
with me, or who have made this effort more delightful, such as Alberto Acerbi,
Anna Jon-And, Anna-Carin Stymne; Arne Jarrick, Elliot Aguilar, Cristina Bicchieri; Fredrik Jansson, Ida Envall, Jan Willem Lindemans; Jiang Ting, Johan
Lind, Laila Nauman; Magnus Enquist, Maria Wallenberg Bondesson, Marijane
Luistro Jonsson; Mark Somos, Markus Jonsson, Micael Ehn; Mícheál de Barra,
Molly Sundberg, Patrik Lindenfors; Sven Isaksson and William English.
Lastly, I am grateful for the support from those that have been close to me.
5
Part I
INTRODUCTION
1
OVERVIEW
This thesis is part of a larger research project on corruption funded by the
Swedish Research Council. It is written within the Ph.D. program in applied
mathematics at Mälardalen University. In this thesis, I apply mathematics to
the domain of corruption and social norms. From a mathematical point of
view, the research is truly applied in the sense that my intended audience is
not specifically mathematicians, but scholars from all disciplines interested in
work on corruption and social norms. These will include scholars from philosophy, economics, sociology, political science, and psychology. Consequently,
my thesis is written with such an audience in mind. It will not have the look
and feel of a typical thesis in mathematics, but there will be no mistaking that
mathematical modeling is what I do.
For mathematical modeling in the area of corruption, game theory is indispensable. After all, corruption has a definite aspect of individuals acting in
narrow self-interest, which is what game theoretic analyses assume. There
are however other aspects, not least social ones. In particular, I will argue
that there are forms of corruption that is in large part norm driven.
Research on corruption, particularily the empirical research, has flourished
in the last few decades. It is typically based on some country measures of
corruption, such as the Corruption Perception Index. Researchers employ
econometric methods to such data to relate corruption to other important
things, such as growth, health, development, and democracy. In this literature,
and often also in the related experimental and theoretic literature, corruption
is often spoken about as if it was a phenomenon that could be covered by a
single theory. Corruption, even if narrowed down to bribery, contain a large
9
1 overview
set of distinct strategic situations. From a potential multi-million dollar bribe
in an arms deal to an average citizen bribing an officer to obtain a driving
license. This thesis interests itself with the latter, the small-scale day-to-day
corruption. The established term for this is petty corruption.
I will start with the premise that petty corruption is largely norm driven,
and that the dynamics of petty corruption fruitfully can be understood as
an example of the dynamics of social norms. The dissertation may, for this
reason, also be read as a text on social norm dynamics.
The thesis is divided into two parts. The first part discusses various fundamental issues: What is corruption? What is known about the process by
which regularities in corrupt behavior can be disturbed? What are norms and
how can they be modeled in mathematical language? What methodologies
are available to modelers of social phenomena? How do the contributions of
the six papers of this thesis relate to the discussion and what do they contribute? The second part of the thesis consists of six research papers. Four of
the papers builds on the background and methodological stances argued for
in the first part. Papers 4 and 5 are of a more psychological nature and considers how the phenomenon at the core of paper 3 is affected by the presence
of value based social groups.
10
2
BACKGROUND
2.1
corruption
Transparency International, a major anti-corruption organization, uses the
following definition of corruption (Eicher, 2012):
“Corruption is the abuse of entrusted power for private gain.”
Other anti-corruption actors, such as the United Nations and the World Bank,
adopt similar ones. This definition is but one example, be it an important one,
another example is: “corruption is a transaction between private and public sector actors through which collective goods are illegitimately converted
into private regarding payoffs” (Heidenheimer et al., 1989). The extension of
either of the two definitions is vast. Including everything from a questionable gift to a preschool teacher to a multi-billion dollar bribe facilitating an
arms deal. The definitions provide ethical categories and help to recognize
violations. If our goal, however, is to map, measure and model corruption,
the broad scope of the definitions pose problems. E.g., how would one make
a questionable gift commensurable to a billion-dollar bribe? Do we expect
the two cases to share the same dynamics? Probably not. Despite these challenges, a large portion of corruption research uses these broad definitions.
Notably, the significant body of econometric work that employ Transparency
International’s Corruption Perception Index, which among other things suggest causes of corruption (Treisman, 2000; Pellegrini, 2011) and correlations
between corruption and growth (Méndez and Sepúlveda, 2006).
To the extent that we lack reasons to believe that distinct types of corruption share causes and effects, these econometric papers are challenging to
11
2 background
interpret. Unsurprisingly, we find more fine-grained taxonomies of corruption. One common differentiation is by type of act, e.g. bribery, embezzlement, fraud, extortion or favoritism (Andvig et al., 2001). It is also common
to contrast grand and petty corruption (Doig and Theobald, 2013). Grand corruption, defined as corruption occurring at the highest levels of government
such that it subverts political, legal and economic systems. Petty corruption,
defined as the smaller-scale corrupt behavior taking place with-in a particular
social framework. In this dissertation, we choose not to explore corruption
in the broad sense, but rather to hone in on petty corruption. Typically in the
form of bribery.
Initially, the economics literature on petty corruption and bribery modeled
the phenomena as situations where the interacting parties have asymmetric
information. E.g. agents do corrupt deals in secrecy and by that know more
about what goes on than their counterparts outside of the deals. This, and
other approaches that stem out of micro economic theory tend to capture
important aspects of the setting, sometimes they, however, prove insufficient.
As Alam (1989) put it:
this [. . . ] perspective on corruption [. . . ] stems from two sources:
the failure to perceive its often systemic character. And an unwillingness to examine its effects in a dynamic perspective.
The systemic character that Alam alludes to, we will assume is explained
largely by social norms. This assumption has empirical support. One illustrative example is the natural cross-cultural experiment in the Fisman and
Miguel (2007) article: Corruption, norms, and legal enforcement: Evidence from
diplomatic parking tickets. It suggests that diplomats from more corrupt countries are more prone to receive and ignore parking ticket. Thus providing
support for our assumption, as corruption levels vary with culture and social
norms. There is a handful of micro level cross-country economic lab experiments supporting the social norms hypothesis too: i) The correlation between
a student’s likelihood to act corrupt and the corruption level of her home
country (Barr and Serra, 2010); ii) The propensity to engage and punish corruption vary with the regional corruption level at the lab location (Cameron
et al., 2009). iii) Similar subject pool effects are also found comparing public
servants and students in Indonesia (Alatas et al., 2009). On the macro level, the
12
2.1 Corruption
social norm assumption about petty corruption is corroborated by the impressive sector correlations of corruption levels in cross-country surveys such as
Transparency International’s Global Corruption Barometer (Riaño et al., 2010).
The cross-cultural variation have received attention before, but the focus
has traditionally been on formal institutions, rather than social norms, e.g. as
in Myrdal (1970):
before the power structure has been changed by evolution or revolution, it will be difficult to decrease corruption or even to hinder
its continual increase.
One notable exception is the Karklins (2005) book: The system made me do it.
It offers an attractive case-study of the importance of social norms to explain
corruption. Depicting how the malign norms of state-avoidance and corruption were fostered by Soviet institutions, and how they spilled over into the
setting of fragile democracies in the 1990s.
Our assumption about social norms being a key determinant for systemic
petty corruption was not least inspired by Bo Rothstein’s so-called “big bang
theory” of anti-corruption (Rothstein, 2007). In summary Rothstein’s paper
holds that systemic petty corruption can only be quenched by a massive resolute institutional overhaul. This as the overhaul would cause citizens to reconsider what is the “way to behave” in relation to the state. Their old behavioral script for interaction, which upheld social norms of corruptive behavior,
would no longer apply. They would, as a result, need to reason about how to
interact, while being aware that the other citizens are reasoning how to behave too. This is important as social norms are coordinative (more about this
in the next section), and a citizen’s change in behavior is conditional on the
other changing too. This rare type of big bang situation could thus facilitate
a new social norms for avoidance of petty corruption.
The purpose of this dissertation is to further the understanding of the dynamics of systemic petty corruption, which we by assumption equate with
the dynamics of social norms.
13
2 background
2.2
social norms
In the previous section, we reduced systemic petty corruption to a social phenomenon determined by social norms. That is, a society may, or may not,
have a social norm to avoid acts of petty corruption. Exploring the dynamics
of systemic petty corruption thus translates to exploring the dynamics of social norm change. This section aims to give some background to what social
norm change have entailed by looking at a few excerpts from the relevant
history of ideas.
The concepts of social norms and conventions have history in all the social
sciences. In both sociology and anthropology the concepts can arguably be
said to be at the very core. To differentiate between the two: Conventions
are informally held understanding with-in a population causing regularities
in social behavior. Social norms, serve a similar purpose but tend to prescribe
a normative behavior in situations where the individual and the social interests do not align. The concepts of social norms and conventions are not only
key, they are old too. Already in ancient Greece social norms (nomos) were
discussed. An example is the “proto-cosmopolitan” Hippas urging the gentlemen gathered to stop quarreling. Arguing that the differences in citizenship
was no more than mere convention (Plato, 1967, Protogoras 337c-d):
Gentlemen, he said, who are here present, I regard you all as kinsmen and intimates and fellow-citizens by nature, not by law: for
like is akin to like by nature, whereas law, despot of mankind,
often constrains us against nature.
In the history of ideas, David Hume has served as the pivotal figure stressing the importance of conventions and norms. Conventions were at the core
of his views on everything from esthetics to justice. Hume defines a convention in Appendix III of the Enquiries Concerning the Human Understanding
and Concerning the Principles of Morals (Hume, 1739/1896, Appendix III):
a sense of common interest; which sense each man feels in his
own breast, which he remarks in his fellows, and which carries
him, in concurrence with others, into a general plan or system of
actions, which tends to public utility
14
2.2 Social norms
Note two things. First, that Humian definition of conventions is closer to
what we today at least colloquially would regard as a social norm. Secondly,
the Humian account is of a psychological nature, in contrast with the formal
definitions that later will serve as the social norm definition throughout most
of this dissertation. The detail of how individuals ponder and feel about norms
do, however, matter. In Paper 3 of the thesis we consider one such case. We
consider the so-called better-than-average bias, the tendency to overestimate
oneself compared to the average person, and how taking this into account
makes escaping a bad social norm or convention even harder.
Conventions are ubiquitous in social life. They give meaning to utterances,
coordinate what side we drive on, and explain how money retains value. They
also are also often proto-social norms, as conventions may become established to the extent that a population develop a liking for the convention and
are ready to police it. To highlight for how long conventions and social norms
have been a topic of study, consider how the idea of money as a convention
has developed through history. Aristotle wrote (Aristotle, 1980, Nicomachean
Ethics, V.5):
Money has become by convention a sort of representative of demand; and this is why it has the name ‘money’ (‘nomisma’) –
because it exists not by nature but by law (nomos) and it is in our
power to change it and make it useless
David Hume picked up the same topic centuries later and explicated the dynamic in On Money (Hume, 1752/1987), as well in the following paragraph
(Hume, 1739/1896, p. 490):
[G]radually establish’d by human conventions without any explicit promise [. . . ] gold and silver become the common measures
of exchange, and are esteem’d sufficient payment for what is of
a hundred times their value.
A century later, the process of becoming conventional was refined further by
economist Carl Menger (1892) in The Origins of Money. The refinement is not
in the language of abstract conventions but suggests by example how the liquid asset of choice changes. I.e. how the convention can be shocked out of an
equilibrium and into a new one by innovation and changes in technology. As
15
2 background
we explore the dynamics of social norms this process of change as a response
to a changing context will be of major importance.
16
3
METHODOLOGY
In this thesis, we will treat the subject matter of social norms and systemic
petty corruption using the language of mathematics. We therefore need to
capture the properties of the phenomena involved and to model them formally. The formal models let us deduce information about the dynamics,
which may later be tested against real world data.
A model is by definition an incomplete description of a phenomenon. What
constitute a good model is an area of dispute. Qualities may be balanced
against one another: Simplicity, predictiveness, fit to historical data, contribution to understanding, etc.. This section does of course not give a full-fledged
overview of the philosophy of social science, but it attempts to summarize
some of the debates and thereby put the papers in context.
3.1
instrumentalism v. realism
A common reaction to the enterprise of modeling social phenomenon is: “How
can people be reduced to numbers?” It is a sensible reaction. The short answer is: They can not. At least not in the sense that the social scientists do.
If we allow us to be less defeatist, it seems clear that we at least seem able to
say something. One way to rescue formal discussion in social science is not
to care about whether models represent any aspects of the social world, but
instead focus solely on its proven predictability. This position is called instrumentalist, and is perhaps best summarized by its poster boy, Milton Friedman,
recipient of the 1976 Nobel Memorial Prize in Economic Sciences (Friedman,
1953, p.14):
17
3 methodology
Truly important and significant hypotheses will be found to have
‘assumptions’ that are wildly inaccurate descriptive representations of reality, and, in general, the more significant the theory,
the more unrealistic the assumptions.
This perspective of what science is, and should do, do not need to claim that
people can be reduced to equations. Instead, the claim is that equations may
be predictive about outcomes of social interaction.
In opposition to the instrumentalists, there is another stance, the realist.
The realists critique the instrumentalism of reducing scholarly activity to the
craft of forecasting. For a realist, a good model should not only be able to
make predictions, but also provide understanding. That is, the realist requires
that the entities within the model represent aspects of the target phenomenon.
Thus, the realist’s perspective may outperform an instrumentalist method if
relevant aspects of the situations suddenly start to change.
Now, how do the realist answer the sensible reaction to “reducing people
to numbers”? One thinkable reply is to claim that she does not do anything of
that sort. That modeling human behavior is to isolate part of what people are,
and then reason about this simpler entity. Paraphrasing Uskali Mäki (2002,
p. 11): The realist’s account is that a model M is a simple system used as a
representation of a more complex system X. The intention is that by studying
the properties and dynamics of M we learn something about X. However,
why would examination of one system, M, convey information about another
system, X? To make it plausible that it does, M should represent X in relevant
aspects.
In this thesis, the modeling intends to commit to a realistic stance in Uskali
Mäki’s sense. A further distinction that one may find apt is Mäki’s ideas of
horizontal and vertical isolation (Mäki, 1992). “Horizontal isolation” entails
methods of abstraction and omission, while “vertical isolation” entail a change
in the level of abstraction. Both kinds of isolation are of course distortions of
any real world representation. That being said, horizontal isolation do allow
us to talk about an isolated dynamics of actual entities in the real world. In
contrast, vertical isolation invents new “aggregation entities” without direct
real world counterpart. Horizontal isolation is thus compatible with an isolationist realistic view of science. In the dissertation, vertical isolation has
18
3.2 Methodological holism v. Individualism
mostly been avoided, but it for instance not honored in the “societal commitment level” construct in Paper 6, the idealization of a shared metric in Paper 1
may also be questioned.
3.2
methodological holism v. individualism
A methodological strife among those not committed to sheer instrumentalism
is what entities a model ought to represent. Is it possible to reduce a complex
social system to behavior among “atoms”? Or, is the only honest representation of the system to consider it as a whole? In social science, these contrasting views are called methodological individualism and methodological
holism respectively.
That is, holism is an anti-reductionist stance, where the whole is purposefully studied in itself. The textbook proponent of the holistic approach is
Émile Durkheim (Durkheim, 1895/1982). Others do too, not least structuralists, such as Claude Lévi-Strauss and Michel Foucault (Lévi-Strauss, 2008; Foucault, 1970/2001).
Methodological individualism, on the other hand, emphasizes that social
phenomena are but patterns in individual behavior, beliefs, and so forth. Individualism is thus a method of horizontal isolation where the “atoms” are
black boxed individuals. This perspective tends to be the one taken in micro
economics. It is also the view taken in relation to social norms by e.g., Max
Weber and Fredrich Hayek (Weber, 1978; Hayek, 1979). In this dissertation, as
a corollary to avoiding vertical isolation, we will commit to methodological
individualism. Note that methodological individualism versus holism is not a
proper dichotomy and that the meaning of the two varies between scientists
(Udehn, 2002).
3.3
simulation v. analytical method
Try to model social norms with assumption true to the real world, and one
will quickly realize that analytical methods are unfeasible. Subjective utility functions, asymmetric discounting factors, and non-trivial interaction networks are only the most well-known aspects that analytical analysis tend to
19
3 methodology
ignore. (Beinhocker, 2007) In recent decades, computer simulations of complex (adaptive) systems have established themselves as an alternative to analytical analysis of mathematical models; opening up analysis of systems that
previously were out of reach. A popular simulation paradigm in social science
is the agent-based modeling (ABM) approach. The approach entails having
agents, representing say individuals, firms or countries. Each agent is given
some, simple or complex, rule of interaction. The simulation then gives the
agents an interaction pattern under which the micro rules may make interesting macro regularities emerge. E.g. segregation, social norms or efficiency
of environments. Note that ABM is compatible with methodological individualism.
The ABM literature is a substantial part of the literature on social norms.
Perhaps most notable is the Robert M. Axelrod (1997) book The complexity
of cooperation: Agent-based models of competition and collaboration which
among other thing illustrate how “social norms” can emerge out of interaction
and learning. Another important contribution is the Joshua M. Epstein (2006)
book Generative social science: Studies in agent-based computational modeling
that consider cases of cultural and social norm change. The Eric D. Beinhocker
(2007) book The Origins of Wealth is another convincing book about the aptitude of ABM. The book is a formidable sales pitch for the methodology, and
how being able to take more of the real-world complex system into account
will benefit science.
I will share my experience with ABM, not as an argument about ABM as a
method, but only as an explanation of why I chose not to extensively use ABM.
My first large ABM-project was to re-implement an agent-based model of
the transition dynamics in corruption (Hammond, 2000). My implementation
could not replicate the findings. I tweaked the implicit assumptions of my
implementations, and all of a sudden it did. Next, I modified the original
model, and after considerable debugging I had an implementation of an ABM
that showed what I had presupposed. However, it left me uncertain to what
degree I should feel that I had corroborated the hypothesis. I ended up feeling
that, for my ends, ABM gave too many degrees of freedom. That said, ABM is
an immensely powerful method. It is not least an excellent tool for exploring
and testing theoretical models, and used with care it can certainly corroborate
theory too.
20
3.3 Simulation v. Analytical method
My limited experience is not a basis for much reasoning, but others have
thought about these issues. First, are simulation and analytical methods qualitatively different epistemologically? The black box nature of simulations,
where we sometimes are left unable to explain why, or upon what assumptions some macro phenomenon hinges, may be considered as such a qualitative difference (Humphreys, 2009). More convincingly, I think, Frigg and
Reiss (2009) argue that the simulation method is “the same old stew”, and that
it is only a difference in degree of what Humpreys calls “epistemically opaqueness”. That said, there are still reasons to think differently about a simulated
corroboration compared to an analytical one, given the typically higher level
of opaqueness.
Proceeding to the challenges I experienced. Till Grüne-Yanoff (2009) argues
that a simulation typically is a weaker corroborator than an analytical result
as the space of possible simulations, especially realistic ones, is far greater
than the number of systems open for analytical analysis. Further, he argues
that the process by which one navigate this space, i.e. implement a simulation
is problematic. Not only do implementing a simulation typically require a set
of implicit assumptions. As the process is one of craftsmanship, the point
at which debugging is discontinued may, as the system is somewhat opaque,
risk being as the expected result is found.
Again, this is not to say that agent-based modeling is not a powerful tool,
it is. Especially for exploration, understanding the implication of a theory, or
in critiquing problematic simplifications (Funcke, 2011).
Through most of the thesis, I opt for analytical methods. In Paper 3 we use
an ABM simulation to illustrate the skewed norm dynamics implied by biased
agents.
21
4
MODELS
4.1
game theory
In this thesis, all papers either use game theory or is motivated by its applications. This section will give a background to the concepts employed.
Game theory is the study of strategic interactions among economic agents
producing outcomes. An economic agent is taken to be an utility-maximizing
rational agent with respect to its preferences. This representation of human
strategic behavior is often dressed up in straw and ridiculed, and sometimes
rightly so. Actual human decision-making is, of course, different. The simplified assumption does, however, provide a normative theory, and seems to
isolate an important part of the underlying dynamics. Also, humans my not
be fully rational, but will in a stable environment over time often tend toward
rational behavior. Not necessarily thanks to individual smarts, but by other
social processes, such as cultural evolution and ecological feedback.
Even with-in the scope of the game theoretic literature there are strains
considering agents with non-perfect rationality. In what has been dubbed evolutionary game theory individuals with fixed (not rationally chosen) strategies
are set to play each other. Individuals are born with the same fixed strategy as
their parents or a random mutation. The individuals with the most successful
strategies will have the greater reproductive success and/or survival. Thus,
their strategy will eventually dominate the population.
Another strain is the so-called bounded rationality literature, it tries to
model and understand how human decision-makers act given their limited
cognitive apparatus. There are many operationalizations, but most common
23
4 models
is to divide reasoning into two systems, a slow, costly rational process and a
toolbox of fast and frugal heuristics employed widely (Gigerenzer and Selten,
2002; Kahneman, 2011). That is, agents are equipped with simple behavioral
rules for classes of situations that they employ without rational questioning.
The game, in game theory, is a description of the strategic situation together with the possible payoff outcomes relative the agent’s preferences.
Each agent chooses a strategy; the combination of all strategic choices of all
the players is a tuple. A Nash equilibrium is a strategy tuple that if selected,
no player have an incentive to individually deviate from it. If the equilibrium
is unique, then this is what the economic agents are predicted to play.
4.1.1 Coordination problems
Coordination games is a class of games in game theory that have multiple
Nash equilibria. In this thesis, all papers are either about coordination problems or motivated thereof. The motivation for why coordination problems
will be central to this thesis will be properly spelled out in Section 4.2. Coordination problems take many forms, let us consider two games, the pure
coordination game, and the stag hunt.
Left
Right
Left
2
0
Right
0
1
Figure 4.1: A pure coordination game example
In Figure 4.1 an example of a pure coordination game is depicted in 2 × 2
normal form game notation. That is, there are two players with a simultaneous choice between two strategies. The game is symmetric; both players may
view the game from the perspective of the row-player. They may thus choose
between the Left and the Right strategy while considering her opponent as
the column-player choosing between the Left and the Right strategy too. The
payoffs are given by the cell indexed by the two player’s strategic choices.
Since coordination games have multiple equilibria, the Nash equilibrium
concept fails to predict what will be the strategies played. Empirically, how-
24
4.1 Game theory
ever, we see that some equilibria are selected more often than other (Sugden,
1989). Payoffs may differ between equilibria, but that fact alone fails to explain differential selection of equilibria. To improve predictiveness, a myriad of Nash equilibrium refinements have been designed to make one of the
Nash equilibria unique (Myerson, 1978; Samuelson, 1998; Govindan and Wilson, 2008). Each refinement is informative for some cases, but no approach
is universal (Fudenberg et al., 1988). Other branches of the literature venture
outside of the specifics of the game in order to make one equilibrium unique.
E.g., formalizing salience (Binmore and Samuelson, 2006; Sugden, 2011), learning (Fudenberg, 1998) or the process of belief formation overall (Perea, 2012;
Kawagoe and Takizawa, 2009).
In pure coordination games, the interests of players align. In the example
in Figure 4.1 both the row and column player prefer to end up in the upperleft equilibrium (Left, Left). In a one-shot interaction, coordination is thus
easily achieved by rational players with rational expectations. Under repeated
interaction, it may be more complicated as players tend to form expectations
using history. If, for one reason or the other, history would be dominated by
play of the inferior lower-right equilibrium (Right, Right) the population may
very well be stuck playing it.
Stag
Hare
Stag
2
0
Hare
1
1
Figure 4.2: A stag hunt game example
Another coordination problem of major importance is the stag hunt (Skyrms,
2004) depicted in Figure 4.2. This is again a symmetric game and both players
will see their strategic situation from the perspective of the row-player. In
this game, agents have a hard time predicting the choice of the other in the
one-shot interaction. The upper-left equilibrium (Stag, Stag) is more risky
while the lower-right equilibrium (Hare, Hare) pays less. It is easy to see that
this is the case, if a player chooses Hare, then she will receive a payoff of 1.
If she instead chooses Stag she will possibly receive a payoff of 2, but only
if her opponent chooses Stag too. If not, she will receive a payoff of 0. The
25
4 models
equilibrium (Hare, Hare) is therefore said to be a risk-dominant equilibrium,
and the equilibrium (Stag, Stag) a payoff-dominant equilibrium.
In essence, what a player needs to do in a coordination problem is to form
a belief about the other’s choice. Rational choice depends on beliefs about
other agent’s beliefs, which makes it hard to uniquely determine a strategic
choice due to indeterminacy of the belief (Bicchieri, 1997, p. 41). A rational
choice analysis of an agent’s choice in a repeated game would thus required to
make strong assumptions about common knowledge concerning rationality,
inductive standards, preferences, background information, etc.. A seemingly
insurmountable task.
4.1.2 Case-based decision making
Game theoretic analysis usually rests on the assumption that agents maximize
expected utility. In Paper 1 we step out of the paradigm of expected utility
theory and instead consider case-based decision theory. Case-based decision
theory was introduced by Gilboa and Schmeidler (2001) as an alternative to
the expected utility theory and rule-based decision-making. As argued above,
expected utility theory is not enough to predict if people will end up in one
rather than the other equilibrium. Case-based decision-making can arguably
be seen as a dynamic version of the rule-based decision-making but without
an a priori carve-up of what situations belong to what rule.
Following Gilboa and Schmeidler (2001), let P be a set of problems, A a set
of acts, and R a set of outcomes. The set of all conceivable cases is then
C ≡ P × A × R.
Case-based decision theory assumes that problems are to varying degrees
similar to each other, captured by a dissimilarity function
s : P × P → [0, 1].
Further, different outcomes are to varying degrees preferred, represented by
a utility function
u : R → R.
26
4.2 Models of petty corruption
A decision maker faced with a problem (p ∈ P) is assumed to base her decision
on her memory of past cases (M ⊂ C) in order to maximize expected utility.
Expected utility will thereby depend on how similar previous cases were to
the problem currently faced:
U(a) = Up,M (a) =
s(p, q)u(r).
(q,a,r)∈M
David Hume might have predated the framework by two and a half century,
but still argues for it well:
“From causes which appear similar we expect similar effects. This
is the sum of all our experimental conclusions.” (Hume, 1748/2000,
p. 31)
4.2
models of petty corruption
In Section 2.1 we went through a variety of perspectives taken on corruption, and in particular on systemic petty corruption. This section will give an
overview of previous modeling pertinent to systemic petty corruption. There
is a vast literature that tie into this problem one way or another. If we, for
instance, consider systemic petty corruption as a phenomenon idealizable as,
say, a classical prisoner’s dilemma or a trust game, then we have shelves upon
shelves of material analyzing these cases. Two prevalent schemes to reach the
“non-corrupt” equilibrium in these settings are either to impose institutions
or to instill social norms. Before moving on with discussing the modeling of
social norms let us take a broader look at previous efforts.
This basic game perspective is far from the only viewpoint, there is a fair
number of work that characterize bribery and corruption as principal-agent
problems (Rose-Ackerman, 1978; Kurer, 1993; Rose-Ackerman, 1999; Bowles
and Garoupa, 1997). That is, as agency-problems, where an agent’s decisions
affect a principal, or where an autonomous single official’s actions affect the
society as a whole. Other interesting perspectives are given by e.g. the Acemoglu and Verdier (1998) paper that introduces a “general equilibrium”-style
model of corruption. The model assumes that the government’s task is to uphold private contracts. The private agents are dishonest and cheat whenever
27
4 models
they get the chance, and government officials are equally dishonest too. The
private agents are divided into “producers” and “suppliers”. The game is one
of asymmetric information. First, the supplier may invest in her intermediate good, which potentially increase the quality. The producer does not know
whether an investment took place or not. Secondly, the producer finds out the
quality of the intermediate good, when he puts the final good on the market.
The supplier, however, will still not know what the quality of her intermediate good was. They contract so that suppliers will reap a benefit from having
provided high-quality intermediate good, and thus have an interest to invest.
In the absence of a contract-upholding government, producers would always
cheat, and thus suppliers would not invest. With a perfect government, the
result is reversed. Acemoglu and Verdier (1998) goes on to introduce various
variations on the incentive structure of the government officials. They may
thereby produce theoretical results concerning salary levels and disciplinary
actions; they also ponder if paying officials too much remove talent from the
private sector.
We choose a game theoretic framework for analysis as it respect methodological individualism and can handle the systemic aspect that we believe is
important. Other types of models do also capture the self-enforcing feedback
loop that make us deem the behavior systemic. One such example is Andvig and Moene (1990) who in a supply and demand setting show how the
profitability of petty corruption may be related to its frequency and suggest
a mechanism for explaining this stylized fact. Hauk and Saez-Marti (2002)
develops a model in a cultural transmission setting, also explaining the stylized fact. Mishra (2006); Basu et al. (1992) are two interesting papers that
hones in on the interaction between different levels of bureaucracy and their
“weakest link of the chain”-dynamics. I.e., having honest officials in a bureaucracy does not help to fight corruption if their supervisors are corrupt. Even
though this is an important aspect, the hierarchical aspects of corruption are
largely omitted from the analysis in this thesis. Other omitted yet interesting aspects include whether corruptees and corruptors require a large portion of trust (Granovetter, 2007), and how occasions for corruption only exist
given corruptor-entrepreneurs (Granovetter, 2007; Coyne et al., 2010; Baumol,
1990).
28
4.3 Models of social norms
4.3
models of social norms
In economics, social norms tend to feature, if at all, as an exogenous factor.
Either as a constraining institution or as an unexplored coordination device.
The literature on sanctions in relation to social norms is however a big topic,
as well as how efficient the social outcomes they create are (Elster, 1989; Wenzel, 2004; Fehr and Fischbacher, 2004).
Another strand of the literature, spearheaded by the David Lewis (1969)
book Conventions, treats conventions and social norms as the subject matter,
rather than as exogenous phenomena. Before coming back to Lewis, let us
acknowledge that for questions about how social norms emerge, other approaches have been more fruitful, such as the simulation work by Axelrod
(1997); Epstein (2006) and the Young (1993) work using Markov processes.
As the title suggests, Conventions is a book that first and foremost is interested in conventions rather than social norms. Lewis do provide a definition
of social norms too, but let us consider his seminal definition of a convention
(Lewis, 1969, p. 76):
A regularity R in the behavior of members of a population P when
they are agents in a recurrent situation S is a convention if and
only if it is true that, and it is common knowledge in P that, in
any instance of S among members of P,
1. everyone conforms to R;
2. everyone expects everyone else to conform to R;
3. everyone has approximately the same preferences regarding all possible combinations of actions;
4. everyone prefers that everyone conform to R, on condition
that at least all but one conform to R;
5. everyone would prefer that everyone conform to R , on condition that at least all but one conform to R ,
where R is some possible regularity in the behavior of members
of P in S, such that no one in any instance of S among members
of P could conform both to R and to R.
29
4 models
He later in the book introduces a more realistic version of the definition
where the specification “almost” is inserted before every instance of “everyone” (Lewis, 1969, p. 78). The definition captures the coordination problem
that Lewis discusses in game theoretic style in his first chapter.
Instead of contrasting his definition of social norms to conventions, let us
instead consider the related definition of social norms from the Cristina Bicchieri (2006) book The Grammar of Society:
Let R be a behavioral rule for situations of type S, where S can
be represented as a mixed-motive game. We say that R is a social
norm in a population P if there exists a sufficiently large subset
Pcf ⊂ P such that, for each individual i ∈ Pcf :
1. Contingency: i knows that a rule R exists and applies to a
situation of type S;
2. Conditional preference: i prefers to conform to R in situations of type S on the condition that (a) and either (b) or (b’)
holds.
(a) Empirical Expectations: i believes that a sufficiently large
subset of P conforms to R in situations of type S;
(b) Normative expectations: i believes that a sufficiently
large subset of P expects i to conform to R in situations
of type S;
(b’) Normative expectations with sanctions: i believes that a
sufficiently large subset of P expects i to conform to R
in situations of type S, prefers i to conform, and may
sanction behavior.
(Bicchieri, 2006, p. 11)
One problem with Lewis’s definition is that a convention is defined per
“recurrent situation”. Trivially there are no such things as recurrent situations. Every situation is unique in at least its extension in time and space.
This suggests that Lewis intended something more like “type of situation”,
which is the concept Bicchieri’s definition explicitly employs. Starting out
with static types of situations may still be problematic though. In Paper 1 we
30
4.3 Models of social norms
show why and generalize the categorization such that social norms are defined per dynamic set of situations. Another aspect of Lewis’s definition that
is limiting, which Bicchieri’s avoids, is the use of common knowledge. Common knowledge implies that something is not only known by everyone, but
also that everyone knows that everyone knows and that this fact is known by
everyone, and so on. This idealization lead to quite different behavior than
more relaxed, and thereby more realistic, assumption of knowledge. The use
of common knowledge in Lewis’s definition has been widely critiqued (e.g.
Binmore (2008)).
In Lewis’s definition, the coordinative nature of the situation under conventions is apparent. In Bicchieri’s definition less so, not least as she talks
about mixed-motive games. The coordinative nature of social norms, however, stem from the condition (a) of her definition, the empirical expectations.
I.e. given a social norm we only want to follow it if we think that a critical
mass of the others will do so.
To think about the definitions, and their game-theoretic implications, let us
consider an example. First, note that social life is rife with conventions and
social norms. Every day most of us just walk through conforming seamlessly.
The equilibrium selection range from the arbitrary, such as on what side of
the road to drive on, to cases where one equilibrium outcome is strongly preferred. Our example is of the latter kind: Imagine a population where everyone prefers a non-littered park to a littered one, even if it means that all (including themselves) must make an effort not to litter. This strategic situation
is assumed to be a mixed-motive game. Agents prefer to be lazy, but would
enjoy a pristine park. Let the payoffs be given by the matrix in Figure 4.3.
This game has a unique equilibrium, to litter (Litter, Litter). The game is an
instance of the well-known prisoner’s dilemma. A social norm may, however,
Litter
Do not litter
Litter
1
3
Do not litter
0
2
Figure 4.3: A prisoner’s dilemma example — styled as littering
make a second equilibrium possible. If there is an anti-littering norm, litter-
31
4 models
ers may be frowned upon, or worse. This transforms the prisoner’s dilemma,
with its unique bad equilibrium, into a stag hunt (recall Figure 4.2). In the
new game (Do not litter, Do not litter) is the payoff-dominant equilibrium, and
(Litter, Litter) is the risk-dominant one.
Now, if an agent would expect the others to keep littering, there is still no
reason for her to make an extra effort not to. Hence, the population may still
be stuck in the risk-dominant equilibria despite the unanimous preference for
the payoff-dominant one. Stated differently, it makes no sense for any one
member of the population to change her behavior unilaterally. Any change
needs to be synchronized.
Not being able to credibly send such a synchronizing signal creates the
hard problem of establishing a new social norm. This can be the case, e.g. if
it is hard to predict what the others will do, or if it is unpractical to convey
to a critical mass of the agents that a critical mass of the others will change
from now on. Most challenging, however, may be the case where what would
be “credible signals” already have been sent, but for some unknown reason
failed to synchronize change. It begs the question: why would it work this
time around?
It is, unfortunately, not uncommon for groups or societies to get stuck in
bad equilibria. Consider for instance parallels to the littering story above
or systemic bribery and kickbacks (covered in the previous section). Another
classic example is the prior non-use of helmets in the National Hockey League
(Schelling, 1978/2006, p. 213). Players did want to wear a helmet, but being
an early adopter made one look weak. When helmets were made mandatory,
most players happily adopting the new standard.
4.4
in summary
Many approaches have been taken to model corruption. In this thesis, we will
only look at a particular type of corruption, namely systemic petty corruption.
This kind of corruption is typically small-scale corrupt behavior taking place
with-in a particular framework of social norms. This lead us to ask questions about how we can shift between frameworks, i.e. how we change social
norms.
32
5
T H E PA P E R S
5.1
on fuzzy norms
Social norms, as defined earlier, are instantiated per type of situation. This
implies that a social norm is isolated from being influenced or influencing behavior in situations of a different type. In the real world, “types of situations”
seem less hermetically sealed against influence. It is not least apparent if we
consider a novel type of situation and agents are forced to guess what is “the
right thing to do”. The guess will almost certainly hinge upon what they do
in similar types of situations.
The purpose of Paper 1 is to alter the Bicchieri (2006) social norm definition; and to make which situations a social norm has influence over dynamic.
To achieve this, we employ the case-based decision framework, letting agents
form belief about “the right thing to do” through a similarity and recencyweighted extrapolation of previous events. The dynamic categorization allows us to consider questions related to norm spillovers between situations.
It also makes norm equilibria escapable and thus norm change, as we see in
the real world, plausible.
Before updating the definition, the paper showcases the dynamics of norm
spillovers in a heavily idealized toy example. The example asks the question:
When is it worthwhile to suspend all but one institution in order to reform
the remaining one, given that we eventually will reintroduce all institutions
as soon as the new behavior is established enough to spill over into them.
33
5 the papers
5.2
on coordination
The obvious candidate scheme for a policy maker to change the prevailing
social norm is to modify the incentive structure. The policy maker thus needs
to ask herself: How much do one need to modify the payoffs to make a critical
mass of the population believe that a critical mass of the population from this
moment onwards will update their behavior?
Classic game theory will not be very informative. The only thing a policy
maker can learn from it is how much she would need to skew incentives in
order to enforce her strategy of choice for any belief held about how others
will play. This typically implies draconian fines.
The need to skew incentives such that the preference ordering holds for all
possible beliefs may seem extreme, but it is not just a theoretical fluke. Suppose that we draw a player randomly from the population, and the payoff is
$1 if both players choose strategy A, and $2 if both choose strategy B. Everyone in this large population has been choosing A for as long as anyone can
remember, and all players are indistinguishable.
Now, suppose that the expected payoff of choosing A in the last two rounds
have decreased by 1 cent per period; say by introduction of surveillance and
policing. Until now, no one has updated their behavior. The expected payoff
for A keeps shrinking. At what point will a critical mass of the population
play B? Before the payoff for A vanishes? Perhaps, but in a large population
of risk-averse members, perhaps not.
The purpose of Paper 2 is not to give the perfect answer to the question
“how big an incentive is required?”. This would require a fruitful theory about
k-level belief formation. Instead, we show how in any coordinative situation
we can introduce a fine and reward mechanism to sidestep the hard problem
while the transfers tend to the infinitesimal as we get close to the benign
equilibrium. We show how this mechanism can be implemented with two
different types of surveillance.
Even though this paper essentially does conceptual exploration it is evident
given our discussion of social norms how the mechanism would relate to its
coordinative game and thus to systemic petty corruption too.
34
5.3 On biases
5.3
on biases
Sometimes belief formation makes us reflect deeply, employing rationality,
history, incentives, signals, and estimations about the others’ belief formation
process and so forth. In coordination games to ge a likelihood distribution of
what a critical mass are likely to do. An introspective process, where one’s
own ideas about the advantages of a certain strategic choice serves as a point
of departure for reasoning. At any step of the belief formation there may be
systematic biases skewing the outcome.
The purpose of Paper 3 is to highlight how such biases interact with social norms and decision making. In particular we will look at a specific wellstudied bias called the better-than-average effect (BAE). BAE is the name for
the tendency we have to overestimate our own good qualities compared to
an average person in the population. For our purposes, imagine a population
stuck in a bad equilibrium. A member of the population may feel that if there
was a good enough chance of re-coordination he would take the chance and
do the “right thing”. He is however biased to believe that he is more prone to
do “the right thing” than the average member of the population, he will thus
underestimate the chance of re-coordination and do nothing. We show that if
this is systematically true in the population then the dynamic will be skewed
such that signals need to be amplified considerably compared to an unbiased
population.
It would be possible to apply a more traditional game theoretic framework
to examine what effect BAE has on the stability of petty corruption. The
problem would then be viewed as a set of two player stag hunt games. In each
of these games the players choose between the corrupt and the non-corrupt
strategy. The outcomes are the same in all games except for the payoffs to
being non-corrupt when the other player is corrupt. This payoff determines
how risky it is to try for the Pareto-optimal outcome. Before deciding on
what to do the players have formed a belief about at which level of risk other
players become corrupt. The players also have a personal risk preference
explicitly modeled since risk domination is a crucial part of a stag hunt game.
These assumptions together with a reasonable updating rule for the belief
formation would lead to the Pareto-optimal equilibrium in less risky games
and the risk dominating in the riskier games. If we now introduce the BAE
35
5 the papers
bias the threshold for how little risk there needs to be in a game to end up
in the non-corrupt outcome shifts. Some games that previously were noncorrupt will now be corrupt due to the change in beliefs. This equilibrium
shift is in line with the effect BAE has in the modeling framework we use in
the paper.
In Paper 4 and Paper 5 we show how BAE specifically may skew the dynamic in more than one direction. The papers empirically suggest how valueladen in-groups actually may reverse the bias. Specifically, one tend to overestimate the average in-group member in comparison to oneself. For our purposes here this showcase how even for well-established general biases the
specifics of the networks, values and social identity can eliminate a bias, or
even turn it on its head.
5.4
on collective actions
Collective action and the dynamics of norms and conventions are related. Not
least when change requires political action. In the case of corruption, we see
how some kleptocracies manage to ramp up corruption levels to full state
capture. In these cases there tend to be no formal democratic process, and for
change to occur, some form of civil uprising may be needed. Less dramatically,
collective action may also model the adoption of a novel social norms, where
a few early adopters need to overcome the collective action problem.
The purpose of Paper 6 is to suggest how we can extend the Mark Granovetter (1978) threshold model to include partial participation. In the new
model, each individual has a de facto threshold per type of action. The per
action threshold is made up of her idiosyncratic threshold, or “willingness to
participate”, and how “effortful” she judge the given action to be for herself.
Instead of the straightforward number of current participation that we get
in Granovetter’s model we here have consider a “societal commitment level”.
This level is the sum of the “signal value” of all actions taken, where the signal
value is the median level of “effortfulness” of an action.
To present the framework we consider the case of public uprisings. We
show that if an instigator can invent ways to participate with any given signal value, then he can create a “stairway of actions”, such that half of the
36
5.4 On collective actions
population (assuming that they initially would participate in some way) will
escalate to full participation. We also show that a dictator may both increase
and decrease the signal value of actions in order to undo an uprising. Collective action is of course also the problem faced by any population keen to
adopt or abandon a social norm.
37
6
F UTURE RESEARCH
Researchers have to make methodological choices; one such choice is where
to land between methodological individualism and methodological holism.
The Lewis (1969) book Conventions sparked a game-theoretical perspective
on norms that since have been used to explore many aspects on how, why
and to what extent social norms are formed. Some aspects of the social norm
dynamics have however been reserved for holistic approaches. The intention
with the case-based decision making in Paper 1 is to suggest a methodological
individualist path forward to be able to explore questions previously reserved
for holistic reasoning. Questions about how established norms change, how
they can co-exist, and how to think about competing norms in the context of
e.g. migration and multiculturalism. The theoretical framework of Paper 1 is
in need of corroboration. Experimental work could help towards this end, and
perhaps also give a firmer grip on the somewhat elusive “similarity” concept.
In Paper 2, we present a mechanism design solution (of sorts) to coordination problems. In doing so, we also make simplifying assumptions that need
to be properly discerned. For example: How stable are the results to the introduction of subjective utilities? To heterogeneous interaction patterns? To
non-perfect inspection? Another interesting question is if real world agents
would trust the mechanism and act in accordance with it, or not. A behavioral
experiment would serve well as a first test.
The take-home message of Paper 3 is that biases do matter in coordinative
dynamics. The better-than-average effect is just one among many possible
systematical biases affecting the social norm dynamics. Extra attention to
how the dynamics may be skewed should thus be spent on social norms con-
39
6 future research
cerning topics related to strong emotional responses, such as disgust, aggression, and sexual attraction.
The last paper sets up a model to suggest the importance of technological
innovation in resolving collective action problems. We concern ourselves in
this paper with the implications for revolutions. There are other interesting
applications of the escalation model, not least in resolving the collective action problem that is facing members of a population trying to establish a new
social norm.
A well-based understanding of how to establish, change and abolish social
norms would greatly aid the fight against corruption in the world today.
40
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