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The Watchman and the Thief - An Experiment on the Comparative Statics in Games. By Dieter Balkenborg and Todd Kaplan. Introduction. Mixed strategy: Randomizing between different options Central role of mixed strategy in games

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The Watchman and the Thief -

An Experiment on the Comparative Statics in Games

By

Dieter Balkenborg

and

Todd Kaplan


Introduction

  • Mixed strategy: Randomizing between different options

  • Central role of mixed strategy in games

  • Mixed strategy as a tool for preventing to be outguessed by the opponent

  • Bart Simpson in episode 9F16 "Good ol' rock. Nuthin' beats that!“. Lisa: “Poor Bart. He is so predictable.”


von Neumann on game theory

Jacob Bronowski recalls in “The Ascent of Man”, BBC, 1973, a conversation with John von Neumann:

… I (JB) naturally said to him, since I am an enthusiastic chess player “You mean the theory of games like chess”.

“No, no,” he (vN) said. “Chess is not a game. Chess is a well-defined form of computation. You may not be able to work out the answers, but in theory there must be a solution, a right procedure in every position. Now real games,” he said, “are not like that at all. Real life is not like that. Real life consists of bluffing, of little tactics of deception, of asking yourself what is the other man going to think I mean to do. And that is what games are about in my theory.”


Mixed Strategies are a way of remaining unpredictable

Penalty Kick:

Dive L

Dive R

58.3

94.97

Kick L

Kick R

92.91

69.92

  • “R” = strong side of kicker

  • Nash prediction for

  • (Kicker, Goalie)=(41.99L+58.01R, 38.54L+61.46R)

  • Actual Data =(42.31L+57.69R, 39.98L+60.02R)

  • Palacios-Huerta (2003), Volij & Palacios-Huerta (2006)


The Watchman and the Thief

WATCHMAN

Q: How does increase in punishment affect equilibrium behaviour?

Watch

Rest

4

High Punishment:

6

Home

5

5

WATCHMAN

THIEF

6

4

4

Rest

Watch

Steal

4

4

7

6

?

Home

5

5

THIEF

4

6

?

Steal

1

7

Low Punishment:

?

?


A: Watchman gets more lazy, theft remains as before.

WATCHMAN

Rest

Watch

4

High Punishment:

6

Home

5

5

WATCHMAN

THIEF

6

4

4

Rest

Watch

Steal

4

4

7

6

Home

5

5

THIEF

4

6

Steal

1

7

Low Punishment:


Game Structures: The Watchman and the Thief

WATCHMAN

Not watching

Watching

0

-1

P >0, punishment for the thief.

F >0, fine for the watchman.

R >0, reward for the watchman.

Not stealing

0

0

THIEF

-F

R

Stealing

1

-P


Mixed Strategy Nash Equilibria

  • Mixed Nash equilibria determined by the condition to make the opponent indifferent

  • How a player randomizes depends only on payoff of his opponent

  • Counterintuitive Implications

  • Relevance for economics: Relevant economic implications (Dasgupta, Stiglitz 1980, Kaplan, Luski, Wettstein 2003) (More firms implies less innovation)


Main design feature

  • Population design (Nagel, Zamir 2001):

  • 10 subjects are row players (thieves)

  • 10 are column players (watchmen)

  • In each period every thief plays against every watchman and vice versa (10 plays)

  • Placement random


Questions

  • Qualitative predictions of theory correct?

  • “Own Payoff effect” to be expected. Will they be wielded out in market-like matching environment?

  • Relevance of security strategies?


  • Literature:

  • Experiments on normal form games with a unique mixed equilibrium

    • Atkinson and Suppes (1963)2x2, 0-sum

    • O'Neill (1987)4x4, 0-sum

    • Malawski (1989)2x2, 3x3 game

    • Brown & Rosenthal (1990)4x4, 0-sum

    • Rapoport & Boebel (1992)5x5

    • Mookerherjee & Sopher (1994)matching pennies

    • Bloomfield (1995)2x2, r. dev,

    • Ochs (1995) 2x2, compet., r.d., r.m.

    • Mookerherjee & Sopher (1997)4x4 games, const sum

    • Erev & Roth (1998)many data sets

    • McKelvey, Palfrey & Weber (1999)2x2

    • Fang-Fang Tang (1999, 2001, 2003) 3x3

    • Rapoport & Almadoss (2000)investment game


  • Literature (cont):

    • Goree, Holt, Palfrey (2000) 2x2 , 0-sum

    • Bracht (2000)2x2 , 0-sum

    • Bracht & Ichimura (2002)2x2, 0-sum

    • Binmore, Swierpinski & Proulx2x2, 3x3, 4x4, 0-sum

    • Shachat (2002)4x4, 5x5 games, 0-sum

    • Shachat & Walker 2x2 games

    • Rosenthal, Shachat & Walker (2001)2x2 games, 0-sum

    • Nagel & Zamir (2000)2x2 games

    • Selten &Chmura (2005)2x2 games

    • Empirical:

    • Walker and Wooders (2001)tennis

    • Ciappori, Levitt, Groseclose (2003)soccer

    • Palacios-Huerta (2003)soccer

    • Volij & Palacios-Huerta (2006)soccer


Outline

  • Experimental Design

  • Aggregate Results

    • Own payoff Effect

    • Quantal Response and Risk Aversion

    • The learning cycle

  • Individual Behaviour

    • Heterogeneity

    • Movers and Shakers

  • Conclusions


Experimental Design:

  • Computerized Experiment at FEELE (the Finance and Economics Experimental Laboratory at Exeter University).

  • Programmed by Tim Miller in z-tree.

  • A session lasted approx. 1:30 h, on average a student earned £18.

  • 120 subjects


  • Written instructions, summary

  • Test questions

  • 2 trial rounds

  • 50 paid rounds

  • Questionnaire, payment

  • No “story”

  • Either row- or column player throughout

  • Each player makes 10 decisions per period


Six Sessions:

  • H1: High penalty

  • H2: High penalty, row interchanged

  • H3: High penalty, column interchanged

  • L4: Low penalty

  • L5: Low penalty, row interchanged

  • L6: Low penalty, column interchanged

    (Data adjusted here)


Screen shot:

Row players’ decision screen


Screen shot:

Column players’ decision screen


Screen shot:

Row players’ “waiting” screen


Screen shot:

Row players’ result screen


Screen shot:

Column players’ result screen


Results 1, Watchmen (col)

  • The average proportion of watching (R)

    is lower in the high-punishment treatments.

  • The observation is robust over time.

  • High Punishment: Too close to 50%?


Results 2, Thieves (row)

average proportion of stealing (B)

  • (Own-payoff effect): In the high-penalty sessions the average proportion of stealing is 10-15% below equilibrium while it is close to equilibrium in the other session.


Results 3

  • The noise in the aggregate per-period data is large. The data fit roughly in a circle of diameter 0.5.

  • There is no convergence to equilibrium.


1

0.9

0.8

1-p

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

q

Session H1


1

0.9

0.8

1-p

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

q

Session H2


1

0.9

0.8

1-p

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

q

Session H3


1

0.9

0.8

1-p

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

q

Session L4


1

0.9

0.8

1-p

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

q

Session L5


1

0.9

0.8

1-p

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

q

Session L6


Quantal Response with Risk Aversion


PENALTY HIGH

WATCH=.28, STEAL=.42

r=0.405808, =0.177009

  • PENALTY=1; (HIGH)

  • WATCH=.28, STEAL=.42

  • r=0.405808, =0.177009

  • WATCH=.32, STEAL=.36

  • r=0.433568, =0.371299

  • WATCH=.34, STEAL=.42

  • r=0.106692, =0.406645

  • PENALTY=4; (LOW)

  • WATCH=.57, STEAL=.48

  • NO CONVERGENCE.

  • WATCH=.65, STEAL=.52

  • r=0.206081, =0.0929088

  • WATCH=.51, STEAL=.52

  • r=1.46235, =0.194742


Results 4

Fraction of watching.

  • The trial rounds matter.

  • The adjustment seems to happen in the first 5 rounds (including trial rounds).


Results 5

  • The ten-period moving averages lie in a circle of radius 0.1.

    Session 6:


Results 6

  • The moving averages data points seem to spin counter-clockwise, although not necessarily around the equilibrium. This is indicated in the following graphs.


Session H1

Direction of movement

Equilibrium


Session H2

Direction of movement

Equilibrium


Session H3

Direction of movement

Equilibrium


Session L4

Direction of movement

Equilibrium


Session L5

Direction of movement

Equilibrium


Direction of movement

Equilibrium

Session L6


Test 1

  • We run a OLS regression on the moving average data using a linear difference equation as the model.

  • Does the matrix describe a rotation counter-clockwise?


Session H1

Graphical Analysis:

Graph exhibiting the DE fitted to the original MA data.The movements are always anticlockwise and converge.

Direction of movement

Equilibrium


Session H2

Graphical Analysis:

Graph exhibiting the DE fitted to the original MA data.The movements are always anticlockwise and converge.

Direction of movement

Equilibrium


Session L3

Graphical Analysis:

Graph exhibiting the DE fitted to the original MA data.The movements are always anticlockwise and converge.

Direction of movement

Equilibrium


Session L4

Graphical Analysis:

Graph exhibiting the DE fitted to the original MA data.The movements are always anticlockwise and converge.

Direction of movement

Equilibrium


Session L5

Graphical Analysis:

Graph exhibiting the DE fitted to the original MA data.The movements are always anticlockwise and converge.

Direction of movement

Equilibrium


Direction of movement

Equilibrium

Session L6

Graphical Analysis:

Graph exhibiting the DE fitted to the original MA data.The movements are always anticlockwise and converge.


Experiment-5

The one-period prediction are quite close.

1 stage Predicted

True Values


Test 2

  • For the smoothed (or original) time series add the angular movements from period to period.

  • The total angle is always positive


Results 7

  • Behaviour of subjects in the same role is very heterogeneous.

  • Many subjects do either not use best replies or are constantly indifferent.

  • However, pure maximin players are rare.

  • Thieves, then watchmen

  • Count bottom and right


H1

L4

H2

L5

L6

H3


H1

L4

L5

H2

H3

L6


Results 8

  • The distribution of aggregate choices tends to have a typical shape, with modes at the ends and a peak around the equilibrium distribution. It is distinctly different from a distribution as on the right.


H1

L4

L5

H2

H3

L6


H1

L4

L5

H2

H3

L6


Results 9

  • Those who played the strategy that was overall the best strategy tended to win the most.


H 1

H 3

H 2

L 6

L 4

L 5

THIEVES’ DECISIONS


EXPERIMENT -1

EXPERIMENT -3

EXPERIMENT -2

EXPERIMENT -6

EXPERIMENT -4

EXPERIMENT -5

WATCHMENS’ DECISIONS


Conclusions

  • Population game like a market?

  • Do anomalies get wielded out?

  • Heterogeneity in behaviour.

  • Still, aggregate behaviour gets close enough to equilibrium for the comparative statics to be true.

  • What explains the own-payoff effect?

  • Repetition with 2 players?

  • A good class room experiment?


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