Pigeons (Columba livia) approach Nash equilibrium in experimental Matching Pennies competitions

Abstract

The game of Matching Pennies (MP), a simplified version of the more popular Rock, Papers, Scissors, schematically represents competitions between organisms with incentives to predict each other's behavior. Optimal performance in iterated MP competitions involves the production of random choice patterns and the detection of nonrandomness in the opponent's choices. The purpose of this study was to replicate systematic deviations from optimal choice observed in humans when playing MP, and to establish whether suboptimal performance was better described by a modified linear learning model or by a more cognitively sophisticated reinforcement-tracking model. Two pairs of pigeons played iterated MP competitions; payoffs for successful choices (e.g., "Rock" vs. "Scissors") varied within experimental sessions and across experimental conditions, and were signaled by visual stimuli. Pigeons' behavior adjusted to payoff matrices; divergences from optimal play were analogous to those usually demonstrated by humans, except for the tendency of pigeons to persist on prior choices. Suboptimal play was well characterized by a linear learning model of the kind widely used to describe human performance. This linear learning model may thus serve as default account of competitive performance against which the imputation of cognitively sophisticated processes can be evaluated.

Keywords: behavioral economics; choice; competition; key peck; mixed-strategy equilibrium; model; pigeons.

Fig. 1

Schematic representation of MP competition. Each of two players (Same and Different) have a choice between heads and tails. Same wins if both players make the same choice; Different wins if players make different choices.

Fig. 2

Choice procedure for unbiasing protocol and MP game. A peck on the center key (illuminated according to block or role) illuminated both side keys. A peck on a side key constituted a choice (left  =  heads, right  =  tails); it changed key color to red and extinguished the opposite side key before delivering food or a blackout.

Fig. 3

Probability of choosing the same alternative 1 to 10 trials in the future, in each of the four games. Probability of repeating a choice was calculated as y  =  p(heads in lag 0|heads in lag x) + p(tails in lag 0|tails in lag x). Fitted curves trace exponential decay functions of the form y  =  y₀ + ae^−x, with y₀ and a varying across games. Fitting was conducted using SigmaPlot 2004 for Windows 9.01.

Fig. 4

Proportion of heads choices across MP games. Pigeon A1 played against B1, and pigeon A2 played against B2. Horizontal bars indicate expected proportion of heads choices according to a fitted Nash Equilibrium model (Equation 3). The ⊕ symbol represents the successful choice with larger reward (top  =  heads, bottom  =  tails). Deviations from horizontal bars towards ⊕ are indicative of own-payoffs effect.

Fig. 5

Mean performance of 1000 simulations of SLLp, with parameters extracted from pigeons' performance. The simulations reproduce reversals in preference across roles predicted by Nash equilibrium, as well as own-payoff effects. Notation is as in Figure 4.

Fig. 6

Proportion of heads choices by pigeon B2 in Game 4 while playing Same, in the first and second half of each session (open and closed circles, respectively). The general trend in change of choice over the game is depicted by a continuous sigmoidal function, y  =  y₀ + (a − y₀) / (1 + e ^{− (x − b) / c}), where y₀, a, b, and c were fitted to choice proportions. The inset shows the difference in choice proportions between the second and first half of each session.