Introduction to Experiments on Game Theoryuctpsc0/Teaching/ECON3020/GT1.pdf · Some Basics in Game Theory I On one hand, games can be divided by the structure of movement: In static

Introduction to Experiments on Game Theory

Syngjoo Choi

Spring 2010

Experimental Economics (ECON3020) Game theory 1 Spring 2010 1 / 23

Game Theory

A game is a mathematical notion of a strategic interaction in whichplayers�payo¤s depend on their own and others�decisions.

sports games such as soccer;bargaining between a �rm and a labor union;bidding for things on eBay, etc.

Game theory is a mathematical tool used to analyze such strategicinteractions and predict people�s behavior.

A game is characterized by

the (number of) players;their sets of feasible actions;the information available at each decision point;the payo¤s as functions of all decisions and random events.


Game Theory and Experimental Method

A usual question: �the theory is interesting ... but do people actuallyplay this way?�

Behaivor in games is notoriously sensitive to details of theenvironment.

To test game theory, we can use data that naturally occur in �eldsettings (such as auctions). But the use of �eld data is often limitedby missing variables on details of strategic environments.

In contrast, experimental control provides a decisive advantage inidentifying the relationship between behavior and environment.


Some Basics in Game Theory I

On one hand, games can be divided by the structure of movement:

In static (or simultaneous-move) games, players simultaneously chooseactions (or do not know others�choices when each of them makes adecision);In dynamic games, players choose actions in some sort of order andobserve the history of the play of the game.

On the other hand, games can be divided by the structure ofinformation:

In games of complete information, each player�s payo¤ function iscommon knowledge among all the players;In games of incomplete information (called Bayesian games), at leastone player is uncertain about another player�s payo¤ function.


Some Basics in Game Theory II

Static games of complete information with 2 players and �nite actionsfor each player are often represented by a matrix form:

A BA u1 (A,A) , u1 (A,B) ,

u2 (A,A) u2 (A,B)B u1 (B,A) , u1 (B,B) ,

u2 (B,A) u2 (B,B)

Each player has two action choices, A or B.

ui (A,B) represents player i�s utility (or payo¤s) when player 1chooses A and player 2 chooses B. For other pairs of actions, theinterpretation is the same.


Some Basics in Game Theory III

A key notion of a game involves a strategy, which is essentially acomplete plan of action that covers all contingencies.

In the previous matrix game, each player�s strategy set is fA,Bg;In either games of incomplete information or dynamic games, astrategy speci�es what a player will do in each contingent situation.

In order to predict which strategies are chosen by players, economistsrely on an equilibrium that speci�es a set of strategies in which eachsingle player has no desire to deviate and choose any other strategy.

John Nash (1950) provided a notion of equilibrium, called Nashequilibrium, in a general environment of strategic interaction.


Dominance

Strategy A strictly (weakly) dominates strategy B if the payo¤ fromchoosing A is higher than (at least as high as) the payo¤ from B, forany strategy choice by other players.

A BA $10, $18,

$14 $6B $4, $7,

$20 $8

This notion of dominance is extremely appealing because A will turnout better than B no matter what you think other players will do.

Thus, it is natural that one choose a dominant strategy even if onedoes not know how rational other players are.


Let�s think about how you will play the following game:

A BA $80, $0,

$80 $100B $100, $35,

$0 $35

This is a so-called Prisoner�s Dilemma game. The dominanceargument predicts that both players choose B, which results in $35for each player. In fact, this is a prediction by Nash equilibrium.


A Prisoner�s Dilemma Experiment

Cooper, DeJong, Forsythe, and Ross (1996, GEB) ran an experimentin which individual subjects were prevented from being rematchedwith the same person:

Here the cooperative player is strategy A. What�s going wrongagainst the equilibrium?(One possible story) Cooperative play (A) can increase the totalwelfare of both players. Players might not be just sel�sh.


A Classroom Experiment

Now let�s play a game in a classroom experiment.

Go to http://veconlab.econ.virginia.edu/login.htm.Session name is ___.


A Classroom Experiment

The game you just participated in is called a guessing game or a�beauty-contest� game, which was �rst experimentally studied byNagel (1995).

A key aspect of most games is the need for players to guess whatothers will do in order to determine their own best decision.

This aspect is somewhat obscured in a 2� 2 game since a wide rangeof beliefs may lead to the same decision.

Nagel�s guessing game with a continuum of decisions, from 0 to 100,may reveal what guess each player make about the average.


Iterated Dominance

Dominance can be applied iteratively.

�rst eliminate dominated strategies for all players;then check whether that �rst round of elimination makes some (initiallyundominated) strategies dominatedeliminate those (iteratedly) dominated strategies, and repeat thisprocess as long as possible.

Games in which this process of iteratively deleting dominatedstrategies leads to a unique equilibrium are called dominance solvable.

In order to do this iterated process, one must believe that others obeydominance as well but it may be less obvious in practice (!).


John M. Keynes�Quotation - Beauty Contest

In his famous book, General Theory of Employment, Interest, andMoney, he draws an analogy between the stock market and anewspaper contest in which people guess what others will guess aremost beautiful (1936, p.156):

"It is not the case of choosing those which, to the best of one�sjudgement, are really the prettiest, nor even those which averageopinion genuinely thinks the prettiest. We have reached the thirddegree, where we devote our intelligences to anticipating what averageopinion expects the average opinion to be. And there are some, Ibelieve, who practise the fourth, �fth, and higher degrees."


Guessing Games

In the guessing games you participated in,

each of N players, i , chooses a number xi in the interval [0, 100]simultaneously;the target number is 0.5 of the average of all chosen numbers,

0.5��

∑Ni=1 xi/N�(in the second game, it is

20+ 0.5��

∑Ni=1 xi/N�);

The player whose number is closest to the target number wins a �xedprize.

The Nash equilibrium (NE) in the �rst game is that all choose 0. Inthe second game the NE is that all choose 40. Why?


NE in Guessing Games

In the �rst game, note that choosing any number in the interval(50, 100] is weakly dominated by 50;

In the �rst step of iterated dominance, each player can eliminate(50, 100].Suppose now each player believes that all others obey dominance andthus the strategy set amounts to [0, 50].In the second step of iterated dominance, each player can eliminate(25, 50] since any number in this interval is dominated by 25.And so on...In�nitely many steps of iterated dominance lead to the unique NE,which is that all players choose 0.

In the second game, the NE is that all players choose 40. (just checkwhether any individual has an incentive to deviate from 40, when allothers choose 40.)


Experimental Results I - Initial Play

Nagel (1995) ran experiments according to the �rst treatment in ourclassroom game.The results for the �rst-period play in her experiments is summarized:


Experimental Results II - Repetition

In Nagel�s experiments subjects replicated the same game four timesand the players chose lower numbers but not 0 at later periods.


Experimental Results III - Newspaper Experiments

Bosch-Domènech, Montalvo, Nagel, and Satorra (American EconomicReview, 2002) ran the same experiments using the advertisement inthree Newspapers: Financial Times (UK), Expansión (Spain), andSpektrum (German).

The target number is (2/3)��

∑Ni=1 xi/N

�.


Comments from Participants in the NewspaperExperiments



S#1206: In case that all numbers are



What Can We Learn From These Experiments?

Sometimes, people�s behavior is not entirely driven by self interest.Other motivations like altruism might be important in economicenvironments.

We will pursue this issue more in Week 4.

Some people think iteratively about what others will do, but this isnot uniform, and the e¤ect of such introspection is not enough tomove decisions to near-Nash levels in a single round of play.

When people are able to learn from experience, behavior doesconverge to Nash equilibrium in this game.

We will pursue the last two issues further in Week 5 and 6.


Documents

Introduction to Experiments on Game Theoryuctpsc0/Teaching/ECON3020/GT1.pdf · Some Basics in Game Theory I On one hand, games can be divided by the structure of movement: In static