Charles University in Prague
Faculty of Social Sciences
Institute of Economic Studies
BACHELOR THESIS
Does Payoff Dominance Matter?
An Experiment.
Author: Michal Polena
Supervisor: PhDr. Lubomı́r Cingl
Academic Year: 2013/2014
Declaration of Authorship
The author hereby declares that he compiled this thesis independently, using
only the listed resources and literature.
The author grants to Charles University permission to reproduce and to distribute copies of this thesis document in whole or in part.
Prague, May 15, 2014
Signature
Acknowledgments
I would like to express my gratitude to my excellent supervisor PhDr. Lubomı́r
Cingl for his numerous helpful comments and advices. He has immensely contributed to my academic development. Moreover, I would like to thank my
mother and my grandparents for their support during the writing of my bachelor thesis as well as during my entire life.
Bibliographic Entry
Polena, M. (2014): ”Does Payoff Dominance matter? An experiment.” (Unpublished bachelor thesis). Charles University in Prague. Supervisor: PhDr.
Lubomı́r Cingl
Length
61 958 characters
Abstract
Risk dominance and Payoff dominance are considered to be the most important
selection criteria in Stag-hunt games. In contrast, the main finding of Schmidt
et al. (2003) is that players do not respond to changes in Payoff dominance
parameter in these games. There might be, however, other explanations for
results of Schmidt et al. (2003). Moreover, Dubois et al. (2011) and Battalio
et al. (2001)’s experimental results suggest that sufficiently large changes in
Payoff dominance parameter may play a role. We, therefore, proposed three
Stag-hunt games in order to examine whether players respond to large changes
in Payoff dominance parameter. Furthermore, we tested the predictive power
of Relative riskiness. Our main finding is that even large changes in Payoff
dominance parameter do not induce players to change their choices. An insignificant trend in players’ choices, caused by Relative riskiness, was detected
in our second finding. Possible explanations are discussed.
JEL Classification
Keywords
C72, C92, D81
Game theory, Stag-hunt game, Selection criteria, Payoff dominance, Relative riskiness
Author’s e-mail
Supervisor’s e-mail
[email protected]
[email protected]
Abstrakt
Risk dominance a Payoff dominance jsou pokládána za nejdůležitějšı́ výběrová
kritéria v hrách lov na jelena. Naproti tomu, hlavnı́ závěry ze studie Schmidt
et al. (2003) jsou, že změny v Payoff dominance parametru neovlivňujı́ hráče
v těchto hrách. Nicméně existujı́ dalšı́ možná vysvětlenı́ pro výsledky studie
Schmidt et al. (2003). Navı́c, experimentálnı́ výsledky Dubois et al. (2011)
a Battalio et al. (2001) naznačujı́, že dostatečně velké změny v Payoff dominance parametru mohou hrát roli. Z tohoto důvodu jsme navrhli tři hry lov
na jelena, abychom prozkoumali, jestli hráči reagujı́ na dostatečně velké změny
v Payoff dominance parametru. Dále jsme otestovali prediktivnı́ sı́lu Relative riskiness. Našı́m hlavnı́m nálezem je, že ani velké změny v Payoff dominance parametru nepřimějı́ hráče ke změně jejich voleb. Nesignifikantnı́ trend
v hráčských volbách způsobený dı́ky Relative riskiness byl objeven v našem
druhém nálezu. V práci jsou diskutována možná vysvětlenı́.
Klasifikace JEL
Klı́čová slova
C72, C92, D81
Teorie her, Lov na jelena hra, Výběrová
kritéria, Payoff dominance, Relative riskiness
E-mail autora
[email protected]
E-mail vedoucı́ho práce [email protected]
Contents
List of Tables
viii
List of Figures
ix
Thesis Proposal
x
1 Introduction
1
2 Game Theory
2.1 A Brief Introduction to Game Theory
2.2 Short Historical Overview . . . . . .
2.3 Game Theory Definitions . . . . . . .
2.4 Classification of Games . . . . . . . .
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3 Stag-hunt Game
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3.1 An Introduction to Stag-hunt games . . . . . . . . . . . . . . . 12
3.2 Stag-hunt Game in the Perspective of Game Theory . . . . . . . 13
3.3 Structure of Stag-hunt Game . . . . . . . . . . . . . . . . . . . 14
4 Selection Criteria & Parameters
4.1 Payoff Dominance Criterion . . . .
4.2 Risk Dominance Criterion . . . . .
4.3 Optimization Premium . . . . . . .
4.4 Relative Riskiness . . . . . . . . . .
4.5 Mixed Strategies & Mixed Minimax
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Regret
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5 Literature Survey
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6 Experiment Design
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Contents
vii
7 Experiment
29
7.1 Experiment Procedure . . . . . . . . . . . . . . . . . . . . . . . 29
7.2 Hypothesis Testing . . . . . . . . . . . . . . . . . . . . . . . . . 30
7.3 Further Data Analyses . . . . . . . . . . . . . . . . . . . . . . . 32
8 Conclusion
35
Bibliography
39
A Instructions
I
B Experimental screen
IV
C Output from Stata
VI
List of Tables
7.1
7.2
7.3
7.4
Demographic Data of Participants
Experimental results . . . . . . .
Participants’ Types . . . . . . . .
Coordination & Miscoordination .
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List of Figures
2.1
2.2
Matching pennies . . . . . . . . . . . . . . . . . . . . . . . . . .
Classification of Stag-hunt game . . . . . . . . . . . . . . . . . .
8
11
3.1
General structure of Stag-hunt games . . . . . . . . . . . . . . .
14
4.1
Regret matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
5.1
5.2
5.3
Schmidt’s et al. Stag-hunt games . . . . . . . . . . . . . . . . .
Battalio’s et al. Stag-hunt games . . . . . . . . . . . . . . . . .
Dubois’s et al. Stag-hunt games . . . . . . . . . . . . . . . . . .
22
24
25
6.1
Experimental Stag-hunt games . . . . . . . . . . . . . . . . . . .
28
A.1 Instruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . II
A.2 Translated instruction . . . . . . . . . . . . . . . . . . . . . . . III
B.1 Game 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IV
B.2 Game 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V
B.3 Game 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V
C.1 Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VI
Bachelor Thesis Proposal
Author
Supervisor
Proposed topic
Michal Polena
PhDr. Lubomı́r Cingl
Does Payoff Dominance Matter? An Experiment.
Topic characteristics Two selection criteria, a Risk dominance criterion and
a Payoff dominance criterion, have been used for explaining outcomes in Staghunt games since they were introduced. Schmidt et al. (2003) find out that
changes in Risk dominance parameter influence players’ choices but changes in
Payoff dominance parameter has no impact on players’ choices. Schmidt et al.
(2003)’s findings may hold true for relatively small changes in Payoff dominance
parameter but the impact of large changes remain unanswered. The aim of this
bachelor thesis is to examine whether large differences in Payoff dominance
parameters have an impact on players’ decisions. An experiment with large
changes in Payoff dominance parameter will be proposed and conducted in
order to answer our research question.
Outline
1. Introduction
2. Literature Review
3. Stag-hunt Game
4. Selection Criteria
5. Experimental Design
6. Results
7. Conclusion
Bachelor Thesis Proposal
xi
Core bibliography
1. Battalio, R., L. Samuelson, & J.V. Huyck (2001): “Optimization incentives and
coordination failure in laboratory stag hunt games.” Econometrica 69(3): pp. 749-764.
2. Carlsson, H. & E. V.Damme (1993): “ Global Games and Equilibrium Selection.”
Econometrica 61 pp. 989-1018
3. Dubois, D., M. Willinger & P. Van Nguyen (2011): “Optimization incentive and
relative riskiness in experimental stag-hunt games.” International Journal of Game
Theory 41(2): pp. 369-380.
4. Feltovich, N., A. Iwasaki & S. H. Oda (2012): “Payoff levels, loss avoidance,
and equilibrium selection in games with multiple equilibria: an experimental study.”
Economic Inquiry 50(4): pp. 932-952.
5. Harsanyi, J. C. (1995): “A new theory of equilibrium selection for games with complete information.” Games and Economic Behavior 8(1): pp. 91-122.
6. Harsanyi, J. C., & R. Selten (1988): “A General Theory of Equilibrium Selection
in Games.” Cambridge: The MIT Press.
7. Rasmusen, E. (2006): “Games and Information: An Introduction to Game Theory.”
Malden: Blackwell Publishing, 4 edition.
8. Rydval, O. & A. Ortmann (2005): “Loss avoidance as selection principle: evidence
from simple stag-hunt games.” Economics Letters 88(1): pp.101-107.
9. Schmidt, D., R. Shupp, J. M. Walker & E. Ostrom (2003): “Playing safe in
coordination games. Games and Economic Behavior.” Games and Economic Behavior
42(2): pp. 281-299.
10. Skyrms, B. (2001): “The Stag Hunt.” Proceedings and Addresses of the American
Phi losophical Association 75(2): pp. 31-41.
Author
Supervisor
Chapter 1
Introduction
A classic dilemma people are confronted with in their everyday lives is the
decision between safe and risky, but potentially more rewarding, choices. Most
of these decisions, such as international conflicts or campaign elections, can be
made only once. In game theory, the dilemma can be modelled by Stag-hunt
game, incorporating both opposing choices. Moreover, one-shot game setting
can represent the decision occurring only once.
Harsanyi & Selten (1988) introduced two selection criteria (Risk dominance
and Payoff dominance criterion) helping players to solve the dilemma. There
has not been a consensus on which of the criteria is more prominent in determining players’ decisions, until Schmidt et al. (2003) closely investigated the
effect of parameters of given selection criteria. Schmidt et al. (2003) found out
that players respond to changes in Risk dominance parameter, but do not respond to changes in Payoff dominance parameter. Nevertheless, Schmidt et al.
(2003) did not control for other parameters of their games, such as Optimization premium and Relative riskiness. In addition, changes in Payoff dominance
parameters in Schmidt et al. (2003)’s games might be considered too small to
induce players to change their decisions. Experimental results of Battalio et al.
(2001) and Dubois et al. (2011) suggest that sufficiently large differences in
Payoff dominance parameter might explain their results.
The purpose of this thesis is to examine whether large changes in Payoff dominance parameter might influence players’ decisions, keeping all other
parameters equal. We proposed, therefore, three Stag-hunt games where we
changed the magnitude of Payoff dominance parameter between Game 1 and
Game 2. Moreover, we changed the magnitude of Relative riskiness parameter
between Game 2 and Game 3 to investigate the predictive power of Relative
1. Introduction
2
riskiness.
Our main finding is that large changes in Payoff dominance parameter do
not influence players’ decisions. Thus, combining our findings with Schmidt
et al. (2003)’s findings, we conclude that players do not respond to any changes
in Payoff dominance parameter at all. Moreover, results of Game 2 and Game
3 show an insignificant effect of Relative riskiness parameter. Potential explanation are provided.
The thesis is organised as follows. In Chapter 2, the terminology of game
theory is defined. The Stag-hunt game is introduced and described in Chapter
3. Chapter 4 is dedicated to selection criteria and parameters used in Staghunt games. Literature review is presented in Chapter 5. Chapter 6 describes
experimental procedure and results. Our main findings are summarized in
Chapter 7.
Chapter 2
Game Theory
2.1
A Brief Introduction to Game Theory
Game theory is a technique used to analyze, model and solve a broad range of
strategic situations in decision making. Different types of strategic situations,
such as conflict situations or cooperative situations, may arise between individuals, groups or institutions every time and everywhere, when one’s payoff is
not solely determined by one’s particular actions but is contingent on decisions
made by other player or players. Therefore, one’s strategic plan of actions must
take expectations of actions taken by others into account. The aim of game
theory is to analyze given strategic situation, model the full range of outcomes
based on one’s and others actions, and then provide one with optimal strategic
plan. The optimal strategic plan is a set of one’s best-respond actions to others
actions earning the highest possible payoff.1
Despite the fact that the notion theory is contained in game theory’s name,
the game theory is not solely theoretical discipline. Game theory is a part
of applied mathematics and has many useful applications in practice. The
study of elections in political science, negotiations in diplomacy, evolutionary
game theory in biology or multi-agent systems in computer science may serve
as examples. The main application of game theory can be, however, found in
economics. The economic phenomena, such as oligopolies, auctions, mechanism
design and general equilibrium, are primarily analyzed by the use of game
theory.
1
Carmichael (2005)
2. Game Theory
2.2
4
Short Historical Overview
Origin of game theory can be traced back to the 17th century, where gambling
was in the background of the development of mathematical probability. Pierre
de Fermat and Blaise Pascal were the first pioneers in the mathematical probability at that time. Further development in mathematics was then used for
studying and analyzing board games, such as chess or draughts. The very first
discussion about game theory was found in the letter of James Waldegrave in
1713 (Bellhouse 2007). In this letter, an optimal strategy for solving a card
game was described.
The Researches into the Mathematical Principles of the Theory of Wealth
was published by Antoine Augustin Cournot in 1838. Duopoly situations in the
markets were studied in this research as well as a solution to these situations
in the form of balanced strategies was provided (Romp 1997). The Researches
into the Mathematical Principles of the Theory of Wealth is the first game
theoretical study in economic science.
The establishment of game theory as a new discipline; however, took place in
the first half of the 20th century. The study ”Zur Theorie der Gessellschaftsspiele”
(in English translated as ”On the Theory of Games of Strategy” (Tucker & Luce
1959) ) written by John von Neumann in 1928 is the foundation stone of game
theory as a discipline. In this study, von Neumann proved a minimax theorem.
The minimax theorem states that every finite, two-person, zero-sum game has
a rational solution in the form of pure or mixed strategy(Poundstone 1992).
The importance of the minimax theorem is demonstrated by von Neumann’s
famous quote: ”As far as I can see, there could be no theory of games ... without that theorem ... I thought there was nothing worth publishing until the
Minimax Theorem was proved”(Casti 1997).
A groundbreaking book ”Theory of Games and Economic Behavior” was
written by John von Neumann together with Oskar Morgenstern in 1944. The
book presented applications of game theory in other disciplines, such as political
science or diplomacy. This fact started an immense development and interest
in game theory in the second half of the 20th century. Moreover, Theory of
Games and Economic Behavior serves as a basis for current game theory.
John Forbes Nash, Reinhard Selten and John Harsanyi were the most influential scholars in game theory in the second half of the 20th century. Nash
proposed an equilibrium of non-cooperative games in which no player has any
incentive to change his or her decision. This equilibrium is nowadays called a
2. Game Theory
5
Nash equilibrium. Selten improved the concept of Nash equilibrium with his
solution principle of subgame perfect equilibria. Harsanyi created new concepts
in game theory, such as complete information and Bayesian games. All three
scholars received together the Nobel Prize in Economic Sciences 1994 for their
contribution to game theory.
In the 21st century, game theory is still developing, and its interdisciplinary
applications are considerably expanding. Two Nobel Prizes in Economic Science have been awarded in this century. The first Nobel Prize was shared by
Thomas Schelling and Robert Aumann in 2005. Schelling and Aumann received the Nobel Prize ”for having enhanced our understanding of conflict and
cooperation through game-theory analysis”2 . Two years later, in 2007, the second Nobel Prize was given to Leonid Hurwicz, together with Eric S. Maskin
and Roger B. Myerson. Hurwicz, Maskin and Myerson become Nobel laureates
”for having laid the foundations of mechanism design theory” 3 .
2.3
Game Theory Definitions
In real life, the rules of a game must be set and described before playing a given
game. This chapter, similarly to the real life approach, will define and describe
rules of games in game theory. These rules will be essential for understanding
the specific features of Stag-hunt game in the next chapter. The following
definitions are based on textbooks of game theory (Gibbons (1992), Rasmusen
(2006) and Carmichael (2005)).
In the first instance, the components of the general game will be defined.
Afterwards, normal-form representation of games will be introduced as a specific form of games. Normal-form games are frequently represented in matrixes
(e.g. Figure 2.1).
2
Nobelprize.
Retrieved from:
sciences/laureates/2005/
3
Nobelprize.
Retrieved from:
sciences/laureates/2007/
http://www.nobelprize.org/nobel prizes/economichttp://www.nobelprize.org/nobel prizes/economic-
2. Game Theory
6
Def.: Elements of a Game
Essential components of a game are:
ˆ a set of players P who make decisions
ˆ an action aij which can be chosen by player i
ˆ a set of all actions Ai = {aij } available to player i
ˆ a player i ’s strategy si which is a plan what action(s) to pick out in a
given moment of a game ; s−i represents strategies of all other players
but player i
ˆ a player i ’s payoff function πi (s1 , . . . , si , . . . , sn ) which shows payoffs de-
pending on player i ’s and others actions
Def.: Normal-Form Representation of Games
Normal-form representation of a game is a description of a game in which each
player simultaneously selects a strategy, and the resulting payoff is determined
as an intersection of chosen strategies.
Player i ’s pure and mixed strategies si determine probability with which a
given action aij from the action set Ai is played.
Def.: Pure Strategy
Pure strategy si is a rule of player i that an action aij is played with probability
one and any other actions from the actions set Ai with probability zero.
Def.: Mixed Strategy
Mixed strategy si is a rule of player i that each action aij from the action set Ai
is played with certain probability pij . It holds that 0 ≤ pij ≤ 1 for j = 1, . . . , N
and pi1 + . . . + piN = 1.
The aim of mixed strategies is to choose such probabilities for given pure
strategies in order to make other player indifferent between her choices. Making other player indifferent between her choices is the best strategy how to
maximize own expected payoff. In addition, in some games, such as Matching
pennies (see figure 2.1), deterministic pure strategies cannot be applied, and
mixed strategies are used instead of pure ones.
2. Game Theory
7
Def.: Best Response
Player i ’s strategy s∗i is called the best response to the strategies s−i selected
by other players if this strategy earns her the greatest possible payoff. Mathematically,
πi (s∗i , s−i ) ≥ πi (s0i , s−i ) ∀s0i 6= s∗i .
If no other strategy is equally good, the best response strategy s∗i is strongly
best. The best response strategy s∗i is weakly best if there is at least one equally
good strategy. (Considering our mathematical entry, the definition of weakly
best response is satisfied with equality.)
Def.: Strictly Dominant Strategy
The strategy s∗i is a called a dominant strategy if it yields higher utility than
all other strategies in all possible states of the world. Formally,
πi (s∗i , s−i ) > πi (s0i , s−i ) ∀s−i ∀s0i 6= s∗i .
Weakly dominant strategy s∗i earns a player i at least as high payoff as any
other strategies. Similarly to the strictly dominant strategy,
πi (s∗i , s−1 ) ≥ πi (s0i , s−1 ) ∀s−i ∀s0i 6= s∗i .
A player is advised by best response what to play given the other players’
actions. Regardless of the other player’ actions, a strictly dominant strategy
recommends an optimal strategy to a player.
Def.: Nash Equilibrium
The mutual combination of players’ best responses (s∗1 , . . . , s∗n ) forms a Nash
equilibrium. No player has an incentive to change his or her strategy in the
Nash equilibrium. Formally,
πi (s∗i , s∗−i ) ≥ πi (s0i , s∗−i ) ∀i ∀s0i .
We can distinguish two types of Nash equilibria: Pure Strategy Nash Equilibrium and Mixed Strategy Nash Equilibrium. When we usually speak about
the Nash equilibrium, we mean the Nash equilibrium in pure strategies. There
are, however, some games, such as Matching pennies (see Figure 2.1), which do
not have any Nash equilibrium in pure strategies. Nevertheless, there must be
2. Game Theory
8
an equilibrium in every finite game (Nash (1950)). Therefore, games without
the Nash equilibrium in pure strategies have the Nash equilibrium in mixed
strategies. Moreover, there are games with multiple Nash equilibria that have
Nash equilibrium in pure strategies as well as in mixed strategies. An example
of a game with both types of equilibria is a Stag-hunt game.
Figure 2.1: Matching pennies
.
Def.: Risk Dominant Nash Equilibrium
Risk Dominant Nash Equilibrium is formed as an intersection of players’ risk
dominant strategies4 .
Def.: Payoff Dominant Nash Equilibrium
In a game with multiple Nash equlibria which are Pareto-rankable 5 , the Payoff dominant Nash Equlibrium is the one which earns a higher payoff to each
player than the other equilibria.
In games with multiple Nash equilibria, such as Stag-hunt games, where
one equilibrium is Risk dominant Nash equilibrium and the other one is Payoff
dominant Nash equilibrium, it is not sure which equilibrium will be played.
Therefore, there are selection criteria and parameters of a game that may predict which Nash equilibrium will be played.
4
Strategies selected by Risk dominance criterion are called risk dominant strategies (See
4.2 Risk Dominance Criterion)
5
The term, Pareto-rankable equilibria, was firstly used by Thomas Schelling in his book
The Strategy of Conflict. This term is used for payoff distinguishable equilibria in games
with multiple Nash equilibria.
2. Game Theory
2.4
9
Classification of Games
Games can be classified according to many different criteria. This section is
particularly concerned with classification criteria which are relevant to the Staghunt game. The classification of games will be helpful for understanding and
describing the Stag-hunt game in the next chapter. According to the Volek
(2010), games can be classified with following classification criteria:
ˆ Number of Players
Games can be classified according to the number of players who play a
given game. We distinguish games with 1, 2 or n players.
ˆ Number of Strategies
There is either a finite number of strategies or infinite number of strategies
in an action set Ai .
ˆ Degree of Allowed Communication
In communication setting, players are allowed to communicate among
themselves. Therefore, players might be likely to reach an agreement
on given actions and to make a binding commitment. Communication
among players can be further discriminated between one-way and two-way
communication. In one-way communication, one player can communicate
her intention to other players. Other players, however, are not allowed
to respond. Thus, the player will not receive any feedback from other
players. On the contrary, in two-way communication players can respond
to each other.
As opposed to communication setting, players are not allowed to communicate among themselves in non-communication games. Hence, players
cannot make any binding commitments.
ˆ Character of Payoffs
Games can be classified according to their sum of all players’ payoffs. We
distinguish constant sum games (antagonistic games) and non-constant
sum games (non-antagonistic games). In constant sum games, a gain of
one player entails a loss of other player because the sum of all players’
payoffs is a given constant 6 . Consequently, the actions of players are in
6
The most well-known type of constant sum game is zero-sum game. Figure 2.1: Matching
pennies is an example of zero-sum game.
2. Game Theory
10
conflict of interest. In non-constant sum games, the player’s actions are
not in conflict of interest and they may cooperate.
ˆ Amount of Information
Players can either have complete or incomplete information in a game.
Complete information means that the structure of the game and the
player’s payoff function is common knowledge among all the players (Gibbons (1992)). On the contrary, incomplete information means that either
information about the structure of game or information about the player’s
payoff function is not known by players.
ˆ Number of Periods
We distinguish one-shot games and repeated games. One-shot games are
played only once. Repeated games are played many times. Repeated
game can be played with fixed number of periods or with some random
number of periods. Random number of periods may be determined with
a device which at the end of every period with some probability decides whether the game will continue or not. Moreover, repeated games
are played with fixed or random matching. Fixed matching means that
player is randomly matched with another player in the first round but she
will stay fixed to that player in the following games. In contrast to the
fixed matching, in random matching players are matched with another
randomly chosen player in every single game.
ˆ Symmetry
Games can be either symmetric or non-symmetric. A symmetric game
means that the choice of a player’s position, whether you are player 1
or player 2, does not change your payoff function. In other words, given
other player’s choice, you will earn the same amount of money irrespective
of your position as player 1 or player 2. On the contrary, your payoff
function in non-symmetric games is dependent on your player’s position.
According to the Bruns (2010), there are only 12 symmetric games out
of all 144 types of distinct 2x2 games.
2. Game Theory
11
The following figure is based on decision-making situation scheme from
Volek (2010). The red circles denote classification features of Stag-hunt games.
Figure 2.2: Classification of Stag-hunt game
Source: Author’s translation of decision-making situation scheme from Volek (2010) .
Chapter 3
Stag-hunt Game
3.1
An Introduction to Stag-hunt games
”If it was a matter of hunting a deer, everyone well realized that he must remain
faithfully at his post; but if a hare happened to pass within the reach of one of
them, we cannot doubt that he would have gone off in pursuit of it without
scruple and, having caught his own prey, he would have cared very little about
having caused his companions to lose theirs.”
{ Jean-Jacques Rousseau }
According to the Skyrms (2001), the Stag-hunt game’s structure as well as
the name came from the above mentioned story. The story about hunting a deer
comes from Discourse on Inequality, book written by Jean-Jacques Rousseau
in 1754. Moreover, Skyrms (2001) claims that examples of Stag-hunt games
can be found in many philosophical works. Two men pulling at the oars of a
boat from A Treatise of Human Nature written by David Hume is one of the
examples.
The Stag-hunt game is considered to be one of the most important games in
game theory because it includes the features and dilemmas of social contracts.
The dilemma between the safe choice and the risky choice with a possible payofffavourable outcome might be found in the business as well as in the real life.
A business example of Stag-hunt game might be two telecommunication firms
offering similar LTE network access. Advertising advantages of LTE 1 over 3G
will raise the consumer’s demand for LTE. If both firms advertise advantages
1
LTE is an abbreviation for Long-Term Evolution. LTE and 3G are communication
networks for high-speed data.
3. Stag-hunt Game
13
of LTE, they will both greatly benefit from it. If one of the firms does not
advertise at all, the firm will earn more than the advertising firm because the
advertising firm has some costs. But the non-advertising firm will benefit less
than in the case when both firms advertise. The real life example of Stag-hunt
game might be the decision of young graduates in Mississippi whether to stay
or not in their country. 2
In game theory, the Stag-hunt game is also known as coordination game,
security dilemma, trust dilemma or assurance game. We would, however, argue
that the name coordination game is an inaccurate appellation for Stag-hunt
game. Furthermore, we claim that the Stag-hunt game should not consider to
be a coordination game at all. According to the Rasmusen (2006), coordination
games are games which share the common feature that the players need to
coordinate on one of the multiple Nash equilibria. In Stag-hunt games, players
wish to coordinate solely on the Payoff dominant equilibrium. If players are in
doubt about coordination on the Payoff dominant equilibrium, they can choose
the risk dominant choice. If players choose the risk dominant choice, they do
not need other player to coordinate on the risk dominant Nash equilibrium. On
the contrary, every player choosing risk dominant choice will be at least weakly
better off if other player selects the payoff dominant choice. This obviously
stands in contradiction with the definition of coordination games.
3.2
Stag-hunt Game in the Perspective of Game
Theory
There is a large number of different types of games in game theory. For the
purpose of this Bachelor thesis, it is, therefore, necessary to clearly classify the
Stag-hunt game with its characteristics . From now on, we will solely consider
games with following characteristics:
ˆ Number of Players
All games are played by two players.
ˆ Number of Strategies
Every player has two pure strategies in her action set Ai .
2
RethinkMississippi. Retrieved from: http://www.rethinkms.org/2014/01/15/mississippianshunt-deer-rabbits-brain-drain-explained-game-theory/
3. Stag-hunt Game
14
ˆ Degree of Communication
Only games without communication are considered.
ˆ Character of Payoffs
Stag hunt games are non-constant sum games.
ˆ Amount of Information
Every player has complete information about the game.
ˆ Number of Periods
One-shot games and first rounds of games with random matching are
solely considered. Schmidt et al. (2003) found that there are no significant differences between one-shot games and first rounds of games with
random matching. Schmidt et al. (2003)’s claim is further supported by
Gallice (2006) and Guyer & Rapoport (1972). It is argued that games
with random matching are similar to the one-shot games. Players cannot
play strategically with respect to the next games as they are rematched
in each period. Moreover, players cannot experience the learning effect
because only the first rounds are considered,
ˆ Symmetry
Every Stag-hunt game is symmetric.
3.3
Structure of Stag-hunt Game
The general structure of Stag-hunt game is depicted in normal-form representation of games in Figure 3.1 .
Figure 3.1: General structure of Stag-hunt games
3. Stag-hunt Game
15
Following the definitions from section 2.3, the general elements of a game
are in our case two players (player 1 and player 2). Each of them has two pure
strategies in her actions set. Player 1 has pure strategies U and V in her action
set A1 . Player 2 has pure strategies L and R in her action set A2 . The last
element of a game is a payoff function represented by the lower and upper case
letters. The upper case letters A, B, C and D are payoffs of player 1. The lower
case letters a, b, c and d are payoffs of player 2. Player’s payoffs are determined
as the intersection of their choices. For example, if player 1 plays U and player
2 plays R, then the intersection of players’ choices is the upper-right matrix
with payoff B for player 1 and payoff c for player 2.
The specific structure of Stag-hunt game is that A > C, D > B, D > A
and B ≥ A. Some games are called Stag-hunt games even though they do not
satisfy B ≥ A. The requirement for B ≥ A, however, is related to the story
about hunting a stag (in Figure 3.1 represented by choices V and R). Less or at
least the same effort is exerted for hunting a hare (in Figure 3.1 represented by
choices U and L) when other player hunts the stag than in situation when other
player hunts the hare too. It should be easier to catch the hare when other
player hunts the stag. Therefore, the payoffs in Stag-hunt game should satisfy
B ≥ A. Preceding structural statements hold for lower case letters in the same
way as for upper case letters because the Stag-hunt game is symmetric.
The best response for player 1 is to play U (V ) when player 2 plays L(R)
because the payoff A(D) is higher than C(B). The same argument holds for
player 2 with his or her payoffs in lower case letters. The Stag-hunt game has
neither strictly nor weakly dominant strategies.
Both players have the same mutual best responses to other player’s decisions. Therefore, there are two Nash equilibria in pure strategies. The first
Nash equilibrium in pure strategies is {U ; L} and the second one is {V ; R}.
According to the Nash equilibrium definition, neither player has an incentive
to deviate from her best response. It holds because A > C and a > c for the
first equilibrium {U ; L} as well as D > B and d > b for the second equilibrium
{V ; R}.
There are two Nash equlibria in Stag-hunt games which yield different payoffs. Thus, the equilibria can be defined as Pareto-rankable equilibria. The
equilibrium {V ; R} is Payoff dominant Nash equilibrium because its payoffs D
for player 1 and d for player 2 are strictly higher than payoffs A and a from
equilibrium {U ; V }. The inferior Nash equilibrium {U ; V } is Risk dominant
Nash equilibrium.
3. Stag-hunt Game
16
The enumeration of Nash equilibria is, however, not complete yet because
there is a remaining equilibrium. It is a Nash equilibrium in mixed strategies.
To maximize one’s own expected payoff in mixed strategies, one should play
her strategies with such probabilities that the other player’s expected payoffs
are the same regardless of her choice. Therefore, the other player is indifferent
between her choices. In other words; considering the general structure of Staghunt game, if player 2 wants to make player 1 indifferent between her choice
U and V , then player 2 must assign probabilities p and 1 − p to her choices L
and R in order to do so.
Player 1’s expected payoffs are Ap+B(1−p) by playing U and Cp+D(1−p)
by playing V . The solution of equation where player 1’s expected payoffs are
equal, formally
Ap + B(1 − p) = Cp + D(1 − p),
provides player 2 with optimal probability p for mixing her choices.
The optimal probability p is
p=
A−C
.
A−C +D−B
A−C
Player 2 should play strategy L with probability p = A−C+D−B
and strateD−B
gy R with 1 − p = A−C+D−B
to make player 1 indifferent between her choices.
Player 1 can make player 2 indifferent in exactly the same way by assigning
probabilities q and (1 − q) to her choices U and V . As the Stag-hunt games
are symmetric, it holds that p = q. The Nash equilibrium in mixed strategies
can be written as
{(U, p =
D−B
)}.
A−C+D−B
A−C
; V, 1
A−C+D−B
−p =
D−B
); (L, q
A−C+D−B
=
A−C
; R, 1
A−C+D−B
−q =
Player’s choices U and L are called risk dominant strategies because payoffs following from playing U or L are being considered safe in comparison
with payoffs achieved by playing choices D or R. Choices D and R are called
pareto dominant strategies because in the case of successful coordination payoff
dominant Nash equilibrium is achieved. Therefore, the trade-off between risk
dominant and pareto dominant strategies constitutes an interesting dilemma
in Stag-hunt games.
Chapter 4
Selection Criteria & Parameters
Selection criteria and particular parameters of Stag-hunt games will be introduced and defined in this chapter. Selection criteria provide us with recommendations which strategy to play in given games. Parameters give us important
information about the payoff structure of games, with suggestion what one
should choose. Both, selection criteria and parameters, are useful for decisionmaking in the presence of multiple Nash equilibria with the trade-off between
risk-safe strategy and risky but pareto efficient strategy. The algebraic notion from section 3.3 Structure of Stag-hunt Game is used for explaining the
selection criteria.
4.1
Payoff Dominance Criterion
”Rational individuals will cooperate in pursuing their common interests if the
conditions permit them to do so . . . ”
{Harsanyi & Selten (1988)}
The above stated quote is the core idea of Payoff dominance criterion.
Harsanyi & Selten (1988) argue in their famous book A General Theory of
Equilibrium Selection in Games that rational players should coordinate on the
Payoff dominant Nash equilibrium. In other words, player 1 and player 2 would
play strategies V and R leading to the most efficient equilibrium with payoffs D
and d. Hence, Harsanyi & Selten (1988) prefer the Payoff dominance criterion
to the Risk dominance criterion (described in the following section). Choices
selected by Payoff dominance criterion are called pareto dominant strategies.
Schmidt et al. (2003) introduced Payoff dominance parameter P as an efficiency loss measurement. Parameter P measures the percentage efficiency
4. Selection Criteria & Parameters
18
loss incurred by successful coordination on the inferior equilibrium in comparison with successful coordination on the Payoff dominant equilibrium. The
parameter P is defined as
P =
p(D, R) − p(U, L)
D−A
d−a
=
=
p(D, R)
D
d
where p(D, R) means payoffs from choice D played by player 1 and from choice
R played by player 2.
4.2
Risk Dominance Criterion
The Risk dominance criterion is the second selection criterion proposed by
Harsanyi & Selten (1988). This selection criterion is based on the comparison
of Nash products of given Nash equilibria. Nash product is a payoff loss of
both players incurred by their deviation from Nash equilibrium. In our case,
the Nash product of Nash equilibrium {U ; L} is (A − C)(a − c) and the Nash
product of Nash equilibrium {V ; R} is (D − B)(d − b). Harsanyi & Selten
(1988) claim that the equilibrium with the higher Nash product should be
chosen. Strategies chosen by Risk dominance criterion are called risk dominant
strategies. In Stag-hunt games, the Payoff dominance criterion and the Risk
dominance criterion have often opposite recommendations. Whilst strategies V
and R are recommended by the Payoff dominance criterion, the Risk dominance
criterion may recommend strategies U and L. The Risk dominance criterion,
however, may recommend payoff dominant strategies V and R too.
For our purposes, we will use slightly modified version of Risk Dominance
Criterion. The modified version of Risk Dominance Criterion measured by
parameter R was proposed by Selten (1995). The parameter R is defined as
R=
A−C
a−c
p(U, L) − p(V, L)
=
=
p(V, R) − p(U, R)
D−B
d−b
where p(U, L) means payoffs (A and a) from intersection of choice U played
by player 1 and from choice L played by player 2. Recommendations given
by the modified vision are the same as recommendations by Nash products.
If parameter R is positive, the strategies, U and L leading to the equilibrium
{U ; L} are recommended. If parameter R is equal to zero, the mixed strategies
are the risk dominant strategies. Strategies V and R leading to the equilibrium
{V ; R} are recommended if parameter R is negative.
4. Selection Criteria & Parameters
4.3
19
Optimization Premium
As opposed to Payoff and Risk dominance criteria, Optimization premium is
rather parameter than selection criterion. The Optimization premium was
invented by Battalio et al. (2001). The Optimization premium parameter, for
our purposes labelled OP, is defined as
OP = [p(U, L)−p(V, L)]+[p(V, R)−p(U, R)] = [A−C]+[D−B] = [a−c]+[b−d].
It is simply the absolute sum of deviation losses from single Nash equilibria
added up together. Battalio et al. (2001) argue that ceteris paribus 1 the higher
the Optimization premium parameter is, the more likely players will choose the
risk dominant strategies.2
4.4
Relative Riskiness
Similarly to the Optimization premium, the Relative riskiness is parameter of
the payoff structure of Stag-hunt games. The Relative riskiness was proposed
by Dubois et al. (2011). The Relative riskiness parameter is labelled RR, and
is defined as
B−A
b−a
p(U, R) − p(U, L)
=
=
.
RR =
p(V, R) − p(V, L)
D−C
d−c
Dubois et al. (2011) argue that ceteris paribus the closer is the Relative riskiness
to zero, the safer the risk dominant strategy seems to be, and players are more
likely to choose the risk dominant strategy. On the contrary, if the Relative
riskiness is close to one, the pareto and risk dominant strategies embrace the
same amount of risk.
There is, however, evident shortcoming in Relative riskiness parameter.
Dubois et al. (2011) assume that payoffs difference p(U, R) − p(U, L) should
not be zero. Otherwise, the value of p(U, R) and p(U, L) would not have any
effect on Relative riskiness. This strong assumption is not always achievable in
Stag-hunt games, which consequently lowers applicability of Relative riskiness.
1
Ceteris paribus means with other conditions remaining the same or other things being
equal.
OxfordDictionaries. Retrieved from: http://www.oxforddictionaries.com/definition/english/
ceteris-paribus
2
From now on, risk dominant strategies refer to players’ strategies leading to the inferior
Nash equilibrium. Using labelling from section 3.3, players’ strategies U and L are considered
to be risk dominant strategies. Reader will be warned when risk dominant strategies are
meant to be pareto dominant strategies.
4. Selection Criteria & Parameters
4.5
20
Mixed Strategies & Mixed Minimax Regret
Mixed strategies are considered to be selection criterion (Gallice (2006)) as
well as particular parameters of Stag-hunt games (Battalio et al. (2001)). The
core idea behind mixed strategies was already explained in Chapter 3. The
inference from Chapter 3 was that players should play strategies U and L
A−C
a−c
with probability pM S = A−C+D−B
= a−c+d−b
to make other player indifferent
between her choices.
Mixed Minimax regret is selection criterion proposed by Gallice (2006).
The computation of the mixed Minimax regret is made in two steps. Firstly,
the common matrix must be altered to the regret matrix. The matrix from
Figure 3.1 was altered into to the regret matrix in Figure 4.1. Gallice (2006)
defines particular payoffs in the regret matrix as regret. The regret is nonnegative difference between the received payoff and the payoff which would be
received if the best response was chosen. For example, the payoffs zero for both
players as intersection of their choices U and L means that originally received
payoffs are equal to the payoffs from best responses. Thus, both players played
optimally, and their regret is equal to zero.
Figure 4.1: Regret matrix
Secondly, the same approach as in Mixed strategies is applied to the regret
matrix. Player 2 properly assigned probabilities to her strategies if player 1’s
expected regrets are equal: p0 + (1 − p)(D − B) = p(A − C) + (1 − p)0. Solution
of the preceding equation is mixed Minimax regret probability:
pM R =
D−B
d−b
=
.
A−C +D−B
a−c+d−b
As the regret matrix is symmetric, both players have the same probability
pM R for playing their strategies U and L. One interesting fact is that probabilities from Mixed strategies and mixed Minimax regret are complementary.
In other words, they add up to one pM S + pM R = 1 (Gallice (2007)).
Chapter 5
Literature Survey
There has not been a consensus on whether players follow Payoff dominance
or Risk dominance criterion since both criteria were introduced in A General
Theory of Equilibria. Harsanyi & Selten (1988), Schelling (1960) and Anderlini
(1999)) believe that player should behave according to the Payoff dominance
criterion and choose the pareto dominant strategy. On the other hand, Carlsson
& Damme (1993), Huyck et al. (1990) and Harsanyi (1995) prefer the Risk
dominance criterion. Although no final conclusion has been drawn yet, there
is prevailing agreement that a Risk dominance criterion is more salient for
outcome prediction (Gallice (2006)). The prevalence of Risk dominance salience
is supported by experimental evidence (Cooper et al. (1992), Straub (1995),
Huyck et al. (1990)) where subjects chose risk dominant strategies more often
than payoff dominant strategies.
Keser & Vogt (2000) proposed and conducted an experiment with a game in
which payoff dominant strategy and risk dominant strategy were identical. In
other words, both selection criteria recommended the same strategy. Therefore,
there should have been a clear outcome predicted by selection criteria. More
than 40% of subjects, however, chose a strategy that was payoff dominated as
well as risk dominated by another strategy. It seems that subjects do not exclusively decide according to the given selection criteria. Parameters of selection
criteria, such as parameter P of Payoff dominance criterion and parameter R
of Risk dominance criterion, should be rather considered. Parameters include
important information about the magnitude of particular selection criterion.
The reason why more than 40% of subjects chose payoff and risk dominated
strategy could be explained by the magnitude of parameter P and R. Moreover, other parameters, such as OP parameter or RR parameter, could play a
5. Literature Survey
22
role.
Schmidt et al. (2003) closer examined the importance and the predicted
power of parameters P and R. Schmidt et al. (2003) proposed four different
Stag-hunt games (depicted in Figure 5.1)1 . Authors argue that the differences
in decisions between games G2 and G3 as well as the differences between games
G1 and G4 are solely attributed to the changes in parameter R because the
parameter P is held constant. Similarly, the differences in decisions between
games G2 and G4 as well as between games G1 and G3 are attributed to the
changes in parameter P while R is held constant. Schmidt et al. (2003) conducted their experiments in three different settings: one-shot games, repeated
games with random matching and repeated games with fixed matching.
Figure 5.1: Schmidt’s et al. Stag-hunt games
Legend: R is Risk dominance parameter. P is Payoff dominance level parameter. OP is
Optimization premium parameter. RR is Relative riskiness parameter. MS stands for Mixed
strategies.
The results from one-shot games and repeated games with random matching
revealed that players respond to the changes in parameter R while parameter
1
The Optimization premium parameter (OP) and the Relative riskiness parameter (RR)
were not included in Schmidt et al. (2003) study. These parameters are added for purposes
of this bachelor thesis.
5. Literature Survey
23
P was kept constant. In one shot games2 , 40% of players played risk dominant
strategies in games G1 and G3. After increasing the value of parameter R
from R = log1 to R = log3, 60% of players in game G2 and 58% of players in
game G4 played the risk dominant strategies. In other words, the change in
parameter R accounts for roughly 19% change in player’s decisions which is a
significant difference according to Schmidt et al. (2003).
On the contrary, the experimental results did not reveal any significant
differences between player’s choices when parameter P was changed, and parameter R was kept constant. In games G1 and G3, 40% of players chose the
risk dominant strategy with parameter P = 0.2 , and the same proportion of
players chose the risk dominant strategy after parameter P was increased to
P = 0.4. Similar results were obtained in games G2 and G4, where 60% of
players chose the risk dominant strategy before and 58% of players after the
parameter P was increased. Considering obtained results, Schmidt et al. (2003)
concluded that players do not respond to the changes in parameter P .
The results of Schmidt et al. (2003), however, do not seem to be in line with
experimental evidence from Battalio et al. (2001) and Dubois et al. (2011).
Moreover, combined experimental evidence from Feltovich et al. (2012) and
Straub (1995) contradicts the Schmidt et al. (2003)’s results as well.
Battalio et al. (2001) proposed and conducted an experiment on three different repeated Stag-hunt games with random matching. (see Figure 5.2)3 . They
wanted to prove the influence of Optimization premium on player’s decisions.
Battalio et al. (2001)’s results are in line with their hypotheses about Optimization premium, but they considered neither parameter P nor parameter
RR in their experiment.
For purposes of this thesis, only results from first periods are reported as a
proxy for one-shot games. Moreover, results are interpreted with the use of the
same parameters as Schmidt et al. (2003) used in their study. All three games
had the same magnitude of parameter R = log4 and increasing magnitude of
parameter P game by game. According to the inferences from Schmidt et al.
(2003), there should not have been any significant difference between games
because parameter R was kept constant. There was no significant difference
2
We decided to demonstrate the Schmidt’s main findings on one-shot games. The inferences from results of repeated games with random matching are, however, the same as in the
one-shot game setting.
3
In line with the arguments stated in section 3.3 Structure of Stag-hunt Games, we do
not consider game 2R to be a proper Stag-hunt game because it does not satisfy A ≥
B. Moreover, the game 2R can be called coordination game because both players have an
incentive to coordinate on one of the Nash equlibria.
5. Literature Survey
24
Figure 5.2: Battalio’s et al. Stag-hunt games
between games R and 0.6R. Nevertheless, there was a significant difference
between games 2R and R. Furthermore, the difference between games 2R and
0,6R was not significant, but the trend that with increasing parameter P players
will be more likely to play payoff dominant strategies was obvious in this case. It
seems that players might not react to small changes in parameter P . In Schmidt
et al. (2003), the parameter P was changed from P = 0.2 to P = 0.4. Thus,
the absolute change in parameter P was only 0.2, which may not be sufficient
cause for players to change their decisions. If parameter P was increased from
P = 0.11 to P = 0.55 or P = 0.733, the significant difference or at least the
trend was detected.
It is necessary to point out that other parameters, such as Optimization
premium parameter (OP ) or Relative riskiness (RR), might have played a
role in determination of Schmidt et al. (2003)’s results. This fact is another
limitation of Schmidt et al. (2003) results because they did not control for other
parameters than Risk dominance parameter, Payoff dominance parameter and
Mixed strategies4 .
Dubois et al. (2011) do not agree with the Battalio et al. (2001)’s inferences
about Optimization premium. Dubois et al. (2011) argue that Battalio et al.
(2001) unconsciously changed Relative riskiness parameter RR while changing
their Optimization premium parameter OP . Dubois et al. (2011) proposed
4
It is not necessary to control for parameter pM S of Mixed strategies because the probability pM S is related to the the parameter R of Risk dominance. Thus, keeping the parameter
R constant in different games means that parameter pM S remains constant too.
5. Literature Survey
25
and conducted an experiment on three repeated Stag-hunt games with random
matching (depicted in Figure 5.3), in which either parameter RR or parameter
OP was changed. Dubois et al. (2011)’s results reject conclusions from Battalio
et al. (2001).
Following the inferences from Schmidt et al. (2003), Dubois et al. (2011)
did not control for Payoff dominance parameter P . The results of Dubois et al.
(2011) could be, however, explained by use of parameter P . Considering the
first rounds of games, there is an obvious trend in results that with an increase
in parameter P players are more likely to choose the payoff dominant strategies.
Even though there is the trend in player’s choices between Games 2 and 3, the
difference between these games is not significant. This results is similar to the
results of Schmidt et al. (2003) and Battalio et al. (2001), because the difference
in Payoff dominance parameter level P is only 0.166 (P3 − P2 = 0.366 − 0.2).
In the same way as in Battalio et al. (2001)’s experiment, we argue that this
difference in parameter P is not sufficient for players to change their choices.
On the other hand, there are significant differences in players’ choices between
Game 1 and Game 2 as well as between Game 1 and Game 3. These differences in players’ choices can be explained by sufficient differences between
Payoff dominance parameters. The difference between parameters P is 0.533
(P1 − P2 = 0.733 − 0.2) in Games 1 and 2, and 0.367 (P1 − P3 = 0.733 − 0.366)
in Games 1 and 3. It seems that players are more likely to choose Payoff dominant strategies in games with sufficiently large Payoff dominance parameter P .
Figure 5.3: Dubois’s et al. Stag-hunt games
Chapter 6
Experiment Design
The main finding of Schmidt et al. (2003) is that Payoff dominance parameter P
does not influence players in selecting their choices in one-shot games as well as
in first rounds of repeated games. The experimental results from Battalio et al.
(2001) and Dubois et al. (2011) indicate that the parameter P might play a role
in determining the outcome. Battalio et al. (2001) explain their experimental
results with use of Optimization premium parameter OP . Dubois et al. (2011)
reject the Battalio et al. (2001)’s results because Battalio et al. (2001) do not
control for the Relative riskiness parameter RR in their experiments. According
to the Dubois et al. (2011), the experimental results of Battalio et al. (2001)
and Dubois et al. (2011) can be explained by use of parameter RR. Neither
Battalio et al. (2001) nor Dubois et al. (2011), however, control for the Payoff
dominance parameter P in their experiments.
The aim of this Bachelor thesis is to closely examine whether the sufficiently
large changes in Payoff dominance parameter P influences players’ choices in
one-shot games. We question the main findings of Schmidt et al. (2003) because
of two reasons. The first reason is that Schmidt et al. (2003) do not control
for Optimization premium parameter OP , and the Relative riskiness parameter
RR is not applicable to the Schmidt et al. (2003)’s games. The second reason is
that the difference in parameters P between given games is only 0.2. We argue
that the difference 0.2 in parameters P is not large enough to induce players to
change their choices. Therefore, the results of Schmidt et al. (2003) may be only
true for small differences in parameters P between given games. Moreover, the
evidence from Battalio et al. (2001) and Dubois et al. (2011) might support
our claim. The differences in players’ choices were not significant in games
with small differences in parameters P . But there were significant differences
6. Experiment Design
27
in players’ choices for large differences in parameters P .
The results of Battalio et al. (2001) cannot be taken as evidence that Payoff
dominance parameter P matters because Battalio et al. (2001) does not control
for parameter P as well as Relative riskiness parameter RR. With the change
of Optimization premium parameter OP , Battalio et al. (2001) change unconsciously the values of parameters P and RR as well. Therefore, it is impossible
to analyse the influence of particular parameters.
Similarly to the Battalio et al. (2001)’s experiment, Dubois et al. (2011) do
not control for parameter P . Thus, Dubois et al. (2011) always change at least
two parameters OP , RR or P at the same time which makes it unfeasible to
analyze effects of particular parameters. One could argue that we do not have
to control for Optimization premium parameter OP because its significance is
rejected. We agree with this argument. Nevertheless, even after omitting the
influence of Optimization premium parameter OP , the Dubois et al. (2011)’s
results cannot be taken as evidence for our hypothesis that parameter P matters. In Dubois et al. (2011)’s Stag-hunt Game 2 and Game 3, the relative
riskiness parameter RR is kept constant but the difference between parameters
P (0.366 − 0.2 = 0.16) in these games is even smaller than in Schmidt et al.
(2003)’s experiment. Thus, these results are not taken into account because the
difference in parameters P is not large enough. The differences in parameters
P are considered to be large enough between Game 1 and Game 2 or between
Game 1 and Game 3. But the Relative riskiness parameter RR is different in
Game 1 in contrast to Game 2 and Game 3. Therefore, it is unclear whether
parameter RR or parameter P influences players’ decision in this case.
We proposed three Stag-hunt games (depicted in Figure 6.1) to closely examine whether the Payoff dominance parameter P does matter or does not,
keeping all other parameters equal. All three games have the same values
of Risk dominance parameter R, Optimization premium parameter OP and
probability Mixed strategy pM S . The differences between games are in use and
magnitude of parameters P and RR. Parameter P is 0.1 in Game 1, and 0.5 in
Game 2 and Game 3. The Relative riskiness parameter RR is 41 in Game 1 and
Game 2, and 12 in Game 3. The differences in players’ choices between Game
1 and Game 2 are solely attributable to changes in parameters P because parameter RR is held constant. The differences in players’ choices between Game
2 and Game 3 are exclusively caused by changes in parameter RR.
6. Experiment Design
28
Figure 6.1: Experimental Stag-hunt games
Hypothesis 1: Significantly more payoff dominant strategies will be chosen
in Game 2 than in Game 1.
We hypothesize that players will be more likely to choose payoff dominant strategies in Game 2 than in Game 1 because of the changes in parameter P . The difference in parameters P between Game 1 and Game 2 is 0.4
(P2 − P1 = 0.5 − 0.1). We conjecture that the difference in parameters P is sufficiently large to induce players to play more often payoff dominant strategies.
Hypothesis 2: No significant differences in players’ choices will be detected between Game 2 and Game 3.
According to Dubois et al. (2011), players should be more likely to play risk
dominant strategies in Game 2 than in Game 3 because of the Relative riskiness parameter RR. Dubois et al. (2011) argue that the closer is the parameter
RR to zero the more likely are players to choose risk dominant strategies. We
challenge the importance of parameter RR because of its limitations in applicability. Moreover, we conjecture that the results of Dubois et al. (2011) and
Battalio et al. (2001) experiments can be rather explained by Payoff dominance
parameter P than by Relative riskiness parameter RR. Therefore, we hypothesize that there will be no significant differences between Game 2 and Game 3
because the value of parameter P is the same for both games.
Chapter 7
Experiment
7.1
Experiment Procedure
The experiment was conducted in the Laboratory of Experimental Economics
(LEE) at the University of Economics in Prague at the beginning of April 2014.
Participants were recruited through ORSEE (Greiner (2004)). Experimental
software z-Tree (Fischbacher (2007)) was used for programming and carrying
out the experiment. In total, three sessions with 68 participants were conducted. The experiment was carried out in Czech. At the beginning of the
experiment, printed instructions were given to participants. Original Czech instruction with their translation into English are enclosed in Appendix A. Each
participant played all three one-shot games in three rounds. Every participant
was matched to another participant after each round. Moreover, participants
did not receive any feedback after each round. Thus, rounds were mutually
independent. To avoid the order effect, there were six different types of order 1 . Only 20 participants out of 24 expected participants came to the first
session. Therefore, the last type of order (G3 → G2 → G1) was not played
in the first session. In the remaining two sessions, all six types of order were
played by 24 participants in each session. At the end of experiment, all payoffs from previous rounds were displayed to participants. One of rounds was
randomly chosen, and players were paid out according to their payoffs from
chosen round. Experimental currency units obtained in Stag-hunt games were
converted to the Czech currency in exchange rate one to one. Participants
earned 43.60CZK(' $2.18) on average. Approximately, two pieces of bread
1
The types of orders were: G1 → G2 → G3 ; G1 → G3 → G2 ; G2 → G1 → G3 ; G2 →
G3 → G1 ; G3 → G1 → G2 ; G3 → G2 → G1.
7. Experiment
30
can be bought for an average payment. The experiment was part of another
experiment, otherwise, the payoffs would be too small for participants. The
experiment was generously funded from the resources of LEE.
Participants were asked to fill out a questionnaire about their demographic
data. The following Table 7.1 summarizes demographic data about participants.
Table 7.1: Demographic Data of Participants
Session
1st
Number of participants
- Man
- Woman
Average age
Degree*
Field of Study
20
13
7
22.5
8
2nd
Total
3rd
24
24
21
18
3
6
22.5
21.9
8
7
Economics or Business
68
52
16
22.3
23
* Degree involves partcipants with bachelor as well as with master degree.
7.2
Hypothesis Testing
Result 1: There is no significant difference in players’ choices between Game
1 and Game 2.
The experimental results do not support our first hypothesis that significantly more payoff dominant strategies (B) will be chosen in Game 2 than in
Game 1. The results from the experiment are summarized in the Table 7.2.
According to the Chi-Square test, the p-value is 0.3. Therefore, the difference
is not significant at p < 0.05.
We infer from our Results 1 that large differences in Payoff dominance
parameter P do not influence players’ choices. Combining our results with existing literature about Payoff dominance parameter P , we conclude that Payoff
dominance parameter P does not play any role in determining the outcome. It
seems that players do not solely respond to the Nash equilibria in Stag-hunt
games as expected by Payoff dominance parameter P .
There are not significantly more payoff dominant choices in Game 2 than
in Game 1 as we hypothesized. But not even a trend towards more payoff
dominant choices in Game 2 is observable. On the contrary, there are more
7. Experiment
31
payoff dominant choices in Game 1 than in Game 2. We conjecture that the
slightly higher number of payoff dominant choices in Game 1 in comparison
with Game 2 is caused by random noise. In order to avoid some discrepancy
in results interpretation, we discuss two other explanations in following paragraphs.
Table 7.2: Experimental results
Game 1
Game 2
Game 3
Decision A
27
33
27
% of A Decisions
39.7%
48.5%
39.7%
Decision B
41
35
41
% of B Decisions
60.3%
51.5%
60.3%
The presence of zero payoff in matrix cells {B; A} for player 1 and {A, B}
for player 2 in Game 2 may be seen as another explanation for the higher
number of payoff dominant choices in Game 1 than in Game 2. This argument
would mean that players are afraid to play payoff dominant strategy B in
Game 2 because it could result in zero payoff. In other words, players would
be considered to be risk averse and choose, therefore, risk dominant choice
A. Nevertheless, Neumann & Vogt (2009) argue that risk averse players are
not more likely to choose risk dominant choice in Stag-hunt games. Thus, the
presence of zero payoff should not induce players to play the risk dominant
choice A.
Considering Figure 7 with proposed Stag-hunt games, one might argue that
the substantial Payoff level difference between Game 1 and Game 2 could play
a role in determining the outcome. This argument may be refuted by Rydval & Ortmann (2005). Rydval & Ortmann (2005) argue that Payoff levels in
Stag-hunt games do not play a role if the payoffs are non-negative. Moreover,
the result from Game 3 supports the Rydval & Ortmann (2005)’s claim. The
number of payoff dominant choices in Game 1 is exactly the same as in Game
3 but Payoff level in Game 1 is more than twice as large as in Game 3.
Result 2: There is no significant difference in players’ choices between
Game 2 and Game 3.
7. Experiment
32
The experimental results support our second hypothesis because no difference between Game 2 and Game 3 was detected at the 5% significance level.
The p-value of the Chi-square test is 0.3. The results from Game 2 and Game
3 are summarized in Table 7.2.
Even though there is no significant difference, we can see higher number
of risk dominant choices A in Game 2 than in Game 3. This point is in
line with Relative riskiness predictions because decreasing the Relative riskiness parameter RR should result in a higher number of risk dominant choices.
Moreover, considering Result 1, it seems that the Payoff dominance parameter
P did not play any role in Battalio et al. (2001) and Dubois et al. (2011)’s
results. Thus, their results should be solely explainable by Relative riskiness.
We should, therefore, observe significant differences in our results if Relative
riskiness truly matters. There are two possible explanations of non-significant
differences.
The first explanation might be that differences in Relative riskiness parameter RR were not large enough to induce players to change their choices. We
increased the parameter RR from 41 in Game 2 to 12 in Game 3. For comparison,
Dubois et al. (2011) increased the parameter RR from 14 in their Game 2 to 23
in their Game 1.
The relatively small number of observations might be seen as another explanation. In total, we have 68 independent observations per game. The number
of our observations is, however, higher than in experiments of Schmidt et al.
(2003) (40 observations per game), Battalio et al. (2001) (64 observations per
game) or Dubois et al. (2011) (64 observations per game). It may suggest that
if there were significant effect of Relative riskiness, it would have been observed
in our experiment.
7.3
Further Data Analyses
In the following paragraphs, we focus on further data analyses of our experimental results. First, the results are regressed on participants’ characteristics,
such as age or obtained degree, in order to determine whether these characteristics can explain the experimental outcome. Different types of participants are
introduced together with their counts in sessions. Differences between sessions
are, further, investigated. Coordination and miscoordination of participants in
different games are examined as well. Next, the types of orders are discussed.
In the end, the gender differences are explored.
7. Experiment
33
We regressed our results from Game 1, Game 2 and Game 3 on participants’
gender, age, obtained degree and number of visited experiments. None of our
independent variables is significant at 5% significance level. The Pseudo R2
ranges from 1.1% in Game 2 to 5.08% in Game 3. We, therefore, conclude
that players’ gender, age, obtained degree and number of visited experiments
do not explain our experimental results well. The output from Stata is enclosed
in Appendix C.
For purposes of our thesis, we define four different types of participants.
Participants who played payoff dominant strategies in all three rounds are called
Payoff Types (P T ). Participans who played once risk dominant strategy are
called rather Payoff Types (rP T ). As Safe Types (ST ) are labeled participants
choosing only risk dominant strategies. Rather Safe Types (rST ) played once
the payoff dominant strategy. The number and percentage of participants in
particular sessions are summarized in Table 7.3.
There are no significant differences in participants’ choices in single games
between all three sessions according to the Fisher’s exact test. The lowest pvalue of 0.358 was obtained as difference between first and second session in
Game 1.
Table 7.3: Participants’ Types
Session
1st
Number
Percentage
2nd Number
Percentage
3rd Number
Percentage
Total Number
Percentage
Participiant’s Type
PT
9
45%
10
41.7%
7
29.2%
26
38.2%
rPT
3
15%
3
12.5%
7
29.2%
13
19.1%
rST
4
20%
4
16.7%
5
20.8%
13
19.1%
ST
4
20%
7
29.2%
5
20.8%
16
23.5%
Participants might either coordinate or miscoordinate (mC) in their choices.
There were two types of coordinations: coordination on a payoff dominant Nash
equilibrium (pC) or coordination on an inferior Nash equilibrium (iC). The
highest number of miscoordinations happened in Game 1. On the contrary,
the lowest number of miscordinations as well as the highest number of payoff dominant coordinations happened in Game 3. In total, participants were
7. Experiment
34
more successful in coordinations in comparison with miscordinations. Table 7.4
summarizes numbers of coordinations and miscoordinations in given games.
The last type of order (G3 → G2 → G1) is significantly different from
third type of order (G2 → G1 → G3) at 10% significance level and from
all other orders at 5% significance level. Moreover, the third type of order
(G2 → G1 → G3) is significantly different from the second type of order
(G1 → G3 → G2) at 10% significance level. Third and sixth type of order
have relatively more payoff dominant choices than other types of orders, and
their common feature is that Game 1 follows after Game 2. Therefore, we
compared experimental results of Game 1 following after Game 2 with results
of Game 1 following after Game 3 and with results of Game 1 as the first game
in an order. We did not find any significant differences in Game 1 under these
three settings. Similarly to the Game 1, Game 2 and Game 3 were compared
in three different settings as well. No significant differences were found.
Table 7.4: Coordination & Miscoordination
Game 1
Game 2
Game 3
Total
Type of coordination
mC
iC
pC
19
4
11
15
9
10
12
7
15
46
20
36
Even though there are no significant differences in gender decision (the
p-value of Fisher’s exact test is 0.3168), the percentage of woman’s payoff
dominant choices is almost 10% higher than by men. Moreover, 50% of women
are Payoff types in comparison with 34.6% of men. The fact that women chose
payoff dominant choices in about 32 of games supports the Neumann & Vogt
(2009)’s statement about risk aversion. As mentioned previously, Neumann &
Vogt (2009) claim that risk aversion of participants does not predetermine them
to choose risk dominant choices. Women are supposed to be more risk averse in
gain domain than men (Eckel & Grossman (2008)) but women were more likely
to choose payoff dominant strategies in our experiment than men. Therefore,
our experimental evidence supports Neumann & Vogt (2009)’s statement.
Chapter 8
Conclusion
To investigate whether large changes in Payoff dominance parameter matter,
two Stag-hunt games were proposed. The Payoff dominance parameter was
greatly increased in Game 2 in comparison with Game 1 while all other parameters were kept equal. Moreover, another Stag-hunt game was proposed
to examine the predictive power of Relative riskiness. Game 3 has the same
values of all parameters except for Relative riskiness parameter as Game 2.
The Relative riskiness parameter in Game 3 is twice as large as in Game 2.
The results from our experiment do not confirm our first hypothesis that
there will be significantly more payoff dominant choices in Game 2 than in
Game 1. In fact, the opposite is true. Participants chose insignificantly more
payoff dominant strategies in Game 1 than in Game 2. We attribute this fact
to the random noise in games. Moreover, two other possible explanations were
discussed.
Considering our results and Schmidt et al. (2003)’s findings, we came to
the conclusion that changes in Payoff dominance parameter P do not have
any significant influence on participants’ decisions. Therefore, the results from
Battalio et al. (2001) and Dubois et al. (2011) cannot be explained by Payoff
dominance parameter.
Our second hypothesis that there will be no significant differences in participants’ choices between Game 2 and Game 3 was confirmed by our experimental
evidence. There is, however, a higher number of payoff dominant choices in
Game 3 in comparison with Game 2. This fact is in line with Relative riskiness
predictions. Moreover, Battalio et al. (2001) and Dubois et al. (2011)’s results
are explainable with Relative riskiness when the influence of Payoff dominance
was rejected. Significant differences should have been observed because of
8. Conclusion
36
the predictive power of Relative riskiness. Two potential explanation for the
insignificant results are provided.
Our recommendation for further research is the development of a generalized
form of Relative riskiness criterion. The current form of Relative riskiness
requires a strong assumption that the payoffs from risk dominant choices are
not equal. This assumption decreases the applicability of Relative riskiness.
For example, in three out of four Schmidt et al. (2003)’s games the Relative
riskiness cannot be applied. We, therefore, suggest the development of the
generalized form applicable to all types of Stag-hunt games.
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Appendix A
Instructions
The original instruction in Czech (Figure A.1) are on the next page followed
by into English translated instruction (Figure A.2).
A. Instructions
II
Figure A.1: Instruction
Instrukce
Tato část experimentu se skládá ze tří na sobě nezávislých kol. V každém kole budete náhodně
přiřazeni k některému z ostatních účastníků experimentu, přičemž Vaše výplata bude záviset na
rozhodnutí tohoto účastníka. V žádné z her nebude hrát s účastníkem, s kterým už jste hráli.
Na obrázku níže vidíte obrazovku, která se Vám zobrazí během experimentu. Prostředí hry se skládá
ze tří polí: levé pole (Vaše výplata), prostřední pole (Vaše volba) a pravé pole (Výplata druhého
účastníka). Písmena W, X, Y a Z označují Vaši možnou výplatu z daného kola v závislosti na rozhodnutí
druhého účastníka a w, x, y a z označují možnou výplatu druhého účastníka v závislosti na Vašem
rozhodnutí. Tato písmena budou během experimentu nahrazena různými číselnými hodnotami,
přičemž W=w, X=x, Y=y a Z=z. Vaše volba (A či B) označuje řádek tabulky, kde se v závislosti na volbě
druhého účastníka určí Vaše výplata. Volba druhého účastníka (A či B) označuje naopak sloupec, kde
v závislosti na Vašem rozhodnutí bude určena jeho výplata.
Příklad
Pokud v prostředním poli zvolíte volbu A a druhý hráč zvolí volbu B, tak Vaše odměna bude X (z
levého pole) a odměna druhého hráče bude y (z pravého pole).
Pokud v prostředním poli zvolíte volbu B a druhý hráč zvolí volbu A, tak Vaše odměna bude Y (z
levého pole) a odměna druhého hráče bude x (z pravého pole).
Na konci experimentu Vám budou zobrazeny Vaše výplaty z těchto tří her. Vyplacena Vám bude
jedna z náhodně vylosovaných her.
A. Instructions
III
Figure A.2: Translated instruction
Instruction
This part of experiment consists of three independent rounds. You will be randomly matched with
one of other participants in each round. Your payoff will be based on your decision as well as on
decision of other participant you will be matched with. You will not be matched twice with the same
participant in any round.
You can see an experimental screen in the picture below. Similar screen will be displayed to you
during the experiment. The game setting is composed of three fields: left field (Your payoff), middle
field (Your choices) and right field (Payoff of other participant). Letters W, X, Y and Z represents your
possible payoffs from a given period depending on your decision and decision of other participant.
Letters w, x, y and z represents the possible payoff of other participant based on his or her decision
and your decision. It holds that W=w, X=x, Y=y a Z=z. The letters will be replaced with numerical
values during the experiment. Your choice (A or B) represents the row from which you will receive
your payoff depending on the other participant choice. The other participant choice (A or B)
represents the column from which you will receive your payoff depending on your decision.
Example
If you select choice A in the middle field and the other participant selects for choice B, then your
payoff will be X (from the left field) and the other participant payoff will be y (from the right field).
If you select for choice B in the middle field and the other participant select for choice A, then your
payoff will be Y (from the left field) and the other participant payoff will be x (from the right field).
All your payoffs from three rounds will be displayed to you at the end of the experiment. One of
randomly selected periods will be paid you off.
Appendix B
Experimental screen
The following three screenshots of Games 1, Game 2 and Game 3 represents
participants’ experimental screens during the experiment.
Figure B.1: Game 1
B. Experimental screen
V
Figure B.2: Game 2
Figure B.3: Game 3
Appendix C
Output from Stata
We regressed our results from Game 1, Game 2 and Game 3 on participants’
gender, age, obtained degree and number of visited experiments. G1-mfx,
G2-mfx and G3-mfx represent marginal effects of particular variables.
Figure C.1: Regression