Giovanna Devetag and Sibilla Di Guida

“Workshop on Rationality, Heuristics and Motivation in Decision Making”

Pisa, November 12-14 2010

Introduction According to traditional game theory, strategic behavior is solely

guided by a game equilibrium structure. Consequently, players’ strategic behavior should not be affected by modifications of a game that leave its equilibrium structure unaltered.

A plethora of experimental studies on single-shot games have shown not only that subjects’ behavior is often out of (Nash) equilibrium, but also that strategizing responds to many features that are theoretically irrelevant (e. g., Bosch-Domènech and Vriend, 2008; Cooper and Van Huyck, 2003; Costa-Gomes et al., 2001; Crawford et al., 2008; Goeree and Holt, 2001, 2004).

New “behavioral” models (k-level model of Costa-Gomes et al. 2001) and equilibrium concepts (QRE, McKelvey and Palfrey, 1995, IBE, Selten and Chmura, 2008, Payoff-sampling equilibrium, Osborne and Rubinstein, 1998; action-sampling equilibrium, Selten and Chmura, 2008) can only partly account for the experimental evidence.

Our hypothesis is that initial behavior in normal form games relies on simplified/incomplete mental models of the strategic situation and hence can be influenced by a set of features that do not alter a game set of (Nash) equilibria.

The presence vs. absence of these features can also influence the extent to which two games are perceived by players as being similar, regardless of their belonging to the same game-theoretic class.

Hence, we: 1)Test whether players’ behavior in normal form games responds to the manipulation of descriptive features

1)Test whether observed behaviors in strategically different games that share the same features are more similar than behaviors in strategically identical games that differ feature wise

Previous literature on single-shot gamesPrevious experimental findings on single-shot games reveal

a high level of heterogeneity, and behavior that is either non-strategic in nature, or strategic in a non-standard sense.

Behavioral models estimated using large data sets (Weizsacker 2003), and experiments that try to track down individual reasoning processes (Devetag and Warglien 2008; Rydval, Ortmann and Ostatnicky 2009) suggest that players have incomplete mental models of the strategic situation, and tend to either ignore their opponents incentives, or to pay attention only to the outcomes in which players’ incentives are perfectly correlated.

The experiment30 3x3 matrices presented in random order60 subjects as row player, 20 subjects in the role of column

players; random matchingControl questionsMost games are not symmetric; our analysis concerns ROW

players only No feedbackPayment on 5 matrices chosen randomly, with random matching

Games and features of interestWe chose the following 5 game types:

1 Dominant strategy for the column player (1 step of it. dom.)2 No pure strategy Nash Equilibria3 Unique pure strategy Nash Equilibrium, not solvable through iterated

elimination of dominated strategies4 Modified Prisoner’s Dilemma5 Modified Weak Link (coordination) game

Features of interest: Presence vs. absence of a Focal Point (FP) Variance of the strategy with Highest Average (HA) payoff for the row

player 6 different versions of each game by varying the presence/absence of

FP, and by introducing 3 levels of variance for the strategy with the highest expected value (HA)

What is a Focal Point? • Our definition of FP differs from the standard one used

in most previous studies. We define a FP as any outcome that • 1) is Pareto efficient, and 2) has symmetric payoffs. It

follows that a FP is not necessarily an equilibrium. We also test the effect of 1) payoff magnitude and 2) cell position in determining the attractiveness of a FP.

Payoff variance as a measure of riskiness• A strategy expected (average) value is important for

“level 1” types of players in k-level models. Nobody has tested the influence of payoff variance in determining the extent to which players exhibit “level 1” type of behavior

We hypothesize that the vast majority of players behave as follows:

1. They choose the strategy with the highest average payoff (HA) when its variance is low orthey choose the strategy supporting the focal point (FP)

As the variance of HA increases, more players choose FP when this is available, and more players choose the equilibrium strategy (EQ)

An example: a game with a dominant strategy for the column player

FP, HA low var FP, HA middle var FP, HA high var

C1 C2 C3   C1 C2 C3   C1 C2 C3  

R1 35,20 35,25 35,30 HA R1 60,20 20,25 25,30 HA R1 80,20 10,25 15,30 HA

R2 5,55 80,80 5,85 FP R2 5,55 80,80 5,85 FP R2 5,55 80,80 5,85 FP

R3 10,20 10,15 40,25 EQ R3 10,20 10,15 40,25 EQ R3 10,20 10,15 40,25 EQ

XFP, HA low var XFP, HA middle var XFP, HA high var

C1 C2 C3   C1 C2 C3   C1 C2 C3  

R1 35,20 35,25 35,30 HA R1 60,20 20,25 25,30 HA R1 80,20 10,25 15,30 HA

R2 5,55 50,25 5,85 XFP R2 5,55 50,25 5,85 XFP R2 5,55 50,25 5,85 XFP

R3 10,20 10,15 40,25 EQ R3 10,20 10,15 40,25 EQ R3 10,20 10,15 40,25 EQ

PD – LOW- FP

C1 C2 C3  

R1 35,10 35,5 35,35

R2 10,35 35,35 5,35

R3 15,15 35,10 10,35

PD – LOW - XFP

C1 C2 C3  

R1 35,10 35,5 35,35

R2 10,35 35,25 5,35

R3 15,15 35,10 10,35

Weak Link

C1 C2 C3  

R1 60,60 35,45 5,35 FP

R2 45,35 45,45 35,35 HA

R3 35,5 35,35 35,35 COS

Weak Link

C1 C2 C3  

R1 35,35 45,45 45,35 HA

R2 5,35 35,45 60,60 XFP

R3 35,35 35,35 35,5 COS

Hypothesis 1 (relevance of FP): For each game type and for each variance level of HA, choice distributions in matrices with FP differ from choice distributions in the corresponding matrices without FP

Hypothesis 2 (relevance of FP and HA over EQ): when variance of HA is low, strategies FP and HA capture the majority of choices in games with FP, and strategy HA captures the majority of choices in games without FP

Hypothesis 3 (effect of variance): Keeping all other features fixed, when the variance of HA increases its share decreases

Hypothesis 4 (nature of focality): the share of the FP strategy increases the more attributes defining a FP are present

Hypothesis 5 (Feature-based weak similarity hypothesis): a feature has a similar effect in different game types, by influencing choice behavior in the same direction

Hypothesis 6 (Feature-based strong similarity hypothesis): keeping all other features fixed, the choice distributions in different game types that are similar with respect to the key feature are closer - statistically – than choice distributions of matrices of the same type that differ with respect to the feature.

Hypothesis 7 (FP response times): matrices without FP are characterized by longer response times than matrices with FP, ceteris paribus.

Analysis of aggregate choices

Hypothesis 1 (relevance of FP): For each game type and for each variance level of HA, choices in matrices with FP differ from choices in the corresponding matrices without FP

Row player Freq. FP Freq. XFPP-value

chi-squareP-value one-tail binomial

DomCol HA low 38% 2% 0.00 0.00

DomCol HA middle 42% 7% 0.00 0.00

DomCol HA high 43% 5% 0.00 0.00

noNE HA low 32% 7% 0.00 0.00

noNE HA middle 50% 7% 0.00 0.00

noNE HA high 58% 0% 0.00 0.00

UniqNE HA low 47% 13% 0.00 0.00

UniqNE HA middle 45% 3% 0.00 0.00

UniqNE HA high 43% 12% 0.00 0.00

PD HA low 10% 5% 0.58 0.24

PD HA middle 17% 5% 0.07 0.04

PD HA high 10% 10% 0.20 0.50

WL HA low 57% 48% 0.60 0.46

WL HA middle 58% 50% 0.62 0.46

WL HA high 82% 77% 0.73 0.65

Hypothesis 2 (relevance of FP and HA over EQ): when variance of HA is low, strategies FP and HA capture the majority of choices in games with FP, and strategy HA captures the majority of choices in games without FP

GameFrequencies of FP + HA low

var in matrices with FPFrequencies of HA with low

var in matrices with XFP

DomCol 83% 80%

noNE 83% 73%

UniqNE 90% 75%

PD 97% 92%

WL 99% 96%

Hypothesis 3 (effect of variance): Keeping all other features fixed, when the variance of HA increases its share decreases

HA low variance

HA middle

variance

HA high variance

Chi-square

test

Binomial test one-

tail

DomCol FP 45% 27% 23% 0.02 0.01

DomCol XFP 80% 48% 43% 0.00 0.00

NoNE FP 52% 37% 20% 0.01 0.00

NoNE XFP 73% 53% 53% 0.00 0.02

UniqNE FP 43% 28% 20% 0.00 0.00

UniqNE XFP 75% 68% 47% 0.00 0.00

PD FP 87% 80% 80% 0.34 0.23

PD XFP 92% 87% 68% 0.00 0.00

Share of the “safe” (constant payoff) strategy

Hypothesis 4 (nature of focality): the share of the FP strategy increases the more attributes defining a FP are present

We identify 4 attributes that may be important in order to increase focality:

payoff magnitude (“significantly” greater than any other payoff the row player can get)

symmetry of payoffs centrality of the cell (or positioned in the main

diagonal in WL) Pareto-efficiency

Result: the joint presence of payoff magnitude and payoff symmetry determines focality

PDDomCol, noNE,

UniqNEWL PD DomCol

Strategy (matrix)

FPlow var

XFPlow var

FPmiddle

var

XFP middle

var

FPlow var

XFP low var

DOM low var

XFP middle

var

Payoff magnitude

X X X X

Symmetry of payoff

X X X X X

Centrality of the cell

X X X X X X

Pareto efficiency

X X X X X X X

Frequency 10% 5% 42% 7% 57% 48% 3% 2%

Hypothesis 5 (Feature-based weak similarity hypothesis): a key feature has a similar effect in different game types, by influencing choice behavior in the same direction

For all game types, the difference in choice shares between matrices with and without features is always significant (p<.01)

The presence of a FP and of a “safe” option (HA) influences choices in predictable ways regardless of a game equilibrium properties.

Hypothesis 6 (Feature-based strong similarity hypothesis): keeping all other features fixed, the choice distributions within-feature type are closer - statistically - than the choice distributions within-game type

For game types DomCol, noNE, and UniqueNAsh in the large majority of cases the frequency distributions are indistinguishable among games sharing the same features.

When the game equilibrium structure changes behavior remains invariant.

The distributions of the game types (except for WL and PD) appear significantly different only when all features are removed.

Analysis of response timesKuo et al. (2009): subjects took much longer, on

average, to choose a strategy in dominance-solvable games as opposed to coordination games, and different areas of the brain activated when players faced instances of the game types. The authors suggest the existence of two different “strategizing” systems in the brain, one based on analytical reasoning and deliberation, the other based on intuition and a “meeting of the minds”

Average response time for each matrix

10,0

15,0

20,0

25,0

30,0

Dom

Col

low

Dom

Col

mid

dle

Dom

Col

hig

h

noN

e lo

w

noN

e m

iddl

e

noN

e hi

gh

Uni

qNE

low

Uni

qNE

mid

dle

Uni

qNE

hig

h

PD

low

PD

mid

dle

PD

hig

h

WL

low

WL

mid

dle

WL

high

Sec

onds

FP XFP

Hypothesis 7 (FP response times): the matrices with FP trigger intuitive reasoning while the matrices without FP trigger analytical reasoning: this difference appears in longer average response times for matrices without FP, ceteris paribus.

RT for matrices with FP are significantly greater than RT for matrices without FP (Wilcoxon signed rank test, p=0.003, one-tailed)

Average response time is (in seconds):17.71 for HA low variance20.98 HA middle variance23.66 HA high variance

Kendall’s W test p=0.000All pairwise differences are statistically significant (p=.00 in all cases)

No sign. correlations were found between individual RT, # of FP choices or # of HA choices.

Positive correlation between individual RT and number of EQ choices (Spearman’s rho coeff. = .331, p=.010, two-tailed) the choice of FP or HA may have derived from players’ imperfect or simplified strategic reasoning rather than from beliefs’ in other players’ irrationality

Equilibrium analysis Nash equilibrium Quantal Response Equilibrium (McKelvey and Palfrey

1995) Action sampling equilibrium (Selten and Chmura, 2008) Payoff sampling equilibrium (Osborne and Rubinstein,

1998) Random choice

Nash equilibrium performs poorly and captures almost none of the effects of the descriptive features. Of all the others stationary concepts analyzed, QRE is the best estimator.

Average of the sum of the squared distances

0,00

0,05

0,10

0,15

0,20

0,25

0,30

0,35

0,40

NashEquilibrium

ActionSampling

Random PayoffSampling

QRE

Strange beliefs or simplified reasoning?An eye-tracking experiment (Devetag&Di Guida, in progress)

Same matricesChoice data confirm previous results

Hypotheses: players who choose HA tend to ignore

opponents’ payoffs

Players who choose FP tend to focus on within-cell comparisons

Own payoff only Infracell payoff

Eye-movements

Eye-movements

Correlations

Number of “Own-payoff by row” movements is positively correlated with HA choices (Spearman’s rho.=0.418, p<.05)

Number of “Infracell” movements is positively correlated with FP choices (Spearman’s rho= 0.365, p<.05)

Players’ behavior in single-shot games can be described by very simple heuristic criteria that eschew optimization.

Different game types that are similar feature-wise are treated equivalently: hence, a feature-based model of cross-game similarity and categorization may be more useful than the standard model to predict behavior in single-shot games

Data on RT and eye-movements suggest that choices of FP and HA do not derive from strange beliefs about opponents but from players’ incomplete representations of the strategic problem

Models based on “types” do not capture feature-based choice

Conclusions