CONTEXTUALITY IN STRATEGIC GAMES

Jerome R. Busemeyer Indiana University

Zheng Joyce WangThe Ohio State University

Supported by AFOSR and NSF

1. Violations marginal invariance for category-decision making task, prisoner dilemma game, and centipede game

2. Quantum cognition explanations violations of marginal invariance

3. Hypothetical example of CBD type contextually for game of chicken.

4. Psychological importance of contextually

ORGANIZATION OF THIS TALK

INTERFERENCE OF CATEGORIZATION ON DECISION

Busemeyer, Wang, Mogiliansky-Lambert (2009, J. of Mathematical Psychology)

Psychological version of a double slit experiment

Wang & Busemeyer(2016, Cognition)

!Categoriza*on,

!

1000/2000ms, 1000ms, 10s,

!Decision,

!

10s,

Feedback,on,C,and,D,

10s,

Participants shown pictures of faces

Categorize as “good” guy or “bad” guyDecide to act “friendly” or “aggressive”

Bad Guys Good Guys

!Categoriza*on,

!

1000/2000ms, 1000ms, 10s,

!Decision,

!

10s,

Feedback,on,C,and,D,

10s,

1000/2000ms, 1000ms, 10s,

!Decision,

!

10s,

Feedback,on,D,

C-then-D: Categorize face first and then decide

D-alone: Decide without categorization

Two Conditions:

• Pr(Bad | Narrow) = .60

• Pr(Good | Wide) = .60

• Pr(Reward Attack | Bad ) = .70

• Pr(Reward Withdraw | Good) = .70

Contingencies ( Learned from Experience)

LAW OF TOTAL PROBABILITY

p(A)= ?=p(G)p(A|G)+ p(B)p(A|B)

D alone Condition C-then-D Condition

G =good guy, B=Bad guy, A=Attack

Total Probability

MARKOV MODEL

Face

Good

Bad

Attack

Withdraw

p(G|F)

p(B|F)

p(A|G)

p(A|B)

P(A|F)=p(G|F)·p(A|G)+p(B|F)·p(A|B)(total probability satisfied)

RG ∩ RW

RG ∩ RA RB ∩ RA

RB ∩ RW

Face

Signal Detection Model

p(A|F) = p(RG ∩ RA|F) + p(RB ∩ RA|F)

= p(RG|F)p(RA|RG,F) + p(RB|F)p(RA|RB,F)

(total probability satisfied)

Face p(G) p(A|G) p(B) p(A|B) TP P(A)

Good 0.84 0.35 0.16 0.52 0.37 0.39

Bad 0.17 0.41 0.82 0.63 0.59 0.69

RESULTS

PRISONER DILEMMA GAME

OD OC

PD O: 10P: 10

O:5P: 25

PC O: 25P: 5

O:20P: 20

Croson, 1999, Organizational Behavior and Human Decision Processes

Game p(c) p(C|c) p(d) p(C|d) TP P(C)

PD-1 0.45 0.83 0.55 0.32 0.55 0.775

PD-2 0.42 0.53 0.58 0.35 0.425 0.625

c = Predict opponent will cooperated = Predict opponent will defectC = Player decides to cooperateTP = total probability cooperate after predictionp(C) = Prob player cooperate with no prediction

Sequentialsocialdilemmagame

Blanco,etal.(2014,GamesandEconomicBehavior)

N = 40 60 60

Centipede6-stagegame,linearincreasingpayoff

EXPERIMENTAL DESIGN

• N = 195 participates paid based on decision outcomes

• Each participant played 30 computer agents Programmed to respond according to base rates from previous research on similar tasks

• All received two types of trials Predict then act Act alone

• For each game/agent, randomly assigned to play first or second Human plays first vs. second

Humanplaysfirst

.731 .93 .684 ..73 .67

.703 .732 .414

Pred-Act

Act-alone

.386

Probability to Continue

Humanplayssecond

.93 .703 .73 .546 .67

.656 .581

Pred-Act

Act-alone

.325

.345

Probability to continue

QUANTUM MODEL OF INTERFERENCE

p(PD) = PPDS2 = PPD ⋅ I ⋅S

2

= PPD ⋅ POD + POC( ) ⋅S 2

= PPD ⋅POD ⋅S + PPD ⋅POC ⋅S2

= PPD ⋅POD ⋅S2 + PPD ⋅POC ⋅S

2 + Int

Int = S | POCPPDPPDPOD | S + S | PODPPDPPDPOC | S

GAME OF CHICKEN

James Dean in “Rebel without

a Cause”

CHICKEN PAYOFF MATRIX

Yield Continue

Yield Chicken, Chicken Chicken, Hero

Continue Hero, Chicken Crash, Crash

Nash equilibrium: choose different action than opponent

HYPOTHETICAL EXPERIMENTAL RESULTS

B-Y B-CA-Y 0 0.5

A-C 0.5 0

2 x 2 contingency tables of relative frequency

C-Y C-C

B-Y 0 0.5

B-C 0.5 0

C-Y C-C

A-Y 0 0.5

A-C 0.5 0

Contextual effects should occur, as expected by traditional game theory

n = 3delta = 0sodd = 3

contextually = sodd - (n-2) - delta = 2 > 0

ADVANTAGE OF QUANTUM MODELS

• Single model that accounts for

• violations of marginal invariance

• order effects

• contextual effects

• Key ingredient required is incompatible (non-commutative) events

CONCLUSIONS

• Violations of marginal invariance can provide strong challenges to major psychological and economic theories

• Contextuality effects can occur, but can (sometimes) be easily explained by existing psychological and economic theories

• Importance of either effect depends on extent to which the effect challenges traditional psychological theory