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Feature-based Choice and Similarity in Normal-form Games: An Experimental Study

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Feature-based Choice and Similarity in Normal-form Games: An Experimental Study

Giovanna Devetag and Sibilla Di Guida

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

Pisa, November 12-14 2010

- 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:

- Test whether players’ behavior in normal form games responds to the manipulation of descriptive features
- 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 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.

- 30 3x3 matrices presented in random order
- 60 subjects as row player, 20 subjects in the role of column players; random matching
- Control questions
- Most games are not symmetric; our analysis concerns ROW players only
- No feedback
- Payment on 5 matrices chosen randomly, with random matching

We chose the following 5 game types:

- Dominant strategy for the column player (1 step of it. dom.)
- No pure strategy Nash Equilibria
- Unique pure strategy Nash Equilibrium, not solvable through iterated elimination of dominated strategies
- Modified Prisoner’s Dilemma
- Modified Weak Link (coordination) game
Features of interest:

- Presence vs. absence of a Focal Point (FP)
- Varianceof 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

- 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.
- 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:

- They choose the strategy with the highest average payoff (HA) when its variance is low
or

they 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)

- 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 FPare 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

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

Share of the “safe” (constant payoff) strategy

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

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.

- Kuo 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”

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.

Responsetimesas a functionof the varof HA

- 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 variance
- 20.98 HA middle variance
- 23.66 HA high variance

- Kendall’s W test p=0.000
- All 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

- 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.

- Same matrices
- Choice 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

Infracell payoff

Own payoff only

- 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