Evolutionary Game Theory in Law and Economics Daria Roithmayr USC Gould School of Law Oct. 29, 2010 DR (Institute) Oct. 29, 2010 1 / 22
Challenges for Neoclassical Game Theory Rationality of Agents: Hyper-rationality not empirically supported. Bounded rationality is more realistic. Equilibrium: Multiple equilibria require Nash refinements, and not every game with pure strategies has a Nash equilibrium. Role of dynamic process: form games, but limited. Agent characteristics: accommodate Some dynamics are possible in extensive Heterogeneity and large n awkward to DR (Institute) Oct. 29, 2010 2 / 22
Evolutionary Game Theory Bounded rationality: Informational asymmetry and uncertainty induces agents to copy strategies from others based on observable payoffs. Equilibrium: Can deal with scenarios that have no equilibrium, and can specify stability with more precision. Dynamics: Can map the game to a system of differential equations to form a dynamical system; can observe dynamic processes like tipping, cycling, etc. Agent characteristics: Heterogeneous populations with large n are incorporated easily. DR (Institute) Oct. 29, 2010 3 / 22
Basics of Evolutionary Game Theory Variation: People choose and use different strategies, by accident, by design, via trial and error. Selection: Environment (in this case, other players) will reward some strategies with higher payoffs. Reproduction: Strategies are "reproduced" via social transmission imitation, learning, etc. DR (Institute) Oct. 29, 2010 4 / 22
Replicator Dynamics How will strategies in population change over time? Are equilibrium points stable? Can we reach these points from any initial conditions? What happens when we perturb the system (displace a variable?) DR (Institute) Oct. 29, 2010 5 / 22
Assumptions in Game Example: Dynamic Hawk Dove H D H...D 1, 1 2, 0 0, 2 1, 2 Games are symmetric. Population is large. Agents are wired to play pure strategies.. DR (Institute) Oct. 29, 2010 6 / 22
Play in Game for Each Period Agents play in pairs. Pairs encounter one another with a probability equal to their frequency in the population. After play, Agents decide whether to switch to strategy of opponent based on relative "fitness" of opponent after play. DR (Institute) Oct. 29, 2010 7 / 22
Dynamic Hawk Dove Game Consider an evolutionary game where each agent follows one of n pure strategies s i for i = 1,..., n. The payoff to s i is π i (p) where i = 1,..., n. Each agent learns with probability 1 the payoff of another randomly chosen agent and switches to opponent strategy with a probability of 1 if the payoff of that strategy is higher than her own. Differential equation: dp i /dt = p i (π i φ) for i = 1,..., n. (1) p i Population frequency of strategy type i π i Payoff of strategy type i φ Weighted average of all payoffs DR (Institute) Oct. 29, 2010 8 / 22
Payoff calculations with Hawk Dove Payoffs from Matrix Payoff of Hawk π h = p h (π hh) + 1 p h )(π hd ) = p h ( 1) + (1 p h )(2) Payoff of Dove π d = p h (π dh) + (1 p h )(π dd ) = p h (0) + (1 p h )(1) Weighted Average of Payoffs φ = p h (π h) + (1 p h )(π d )) = p h (p h ( 1) + (1 p h )(2) + (1 p h )(p h (0) + (1 p h )) DR (Institute) Oct. 29, 2010 9 / 22
Replicator Dynamics Equation with Hawk Dove Payoffs Plugging in payoff values from matrix, we can rewrite the differential replicator equation (1) as: dp h /dt = p h (1 p h )(1 2p h ) Note: p h (1 p h ) is a term that represents variance, and this creates the selection pressure that drives the dynamic movement of the system DR (Institute) Oct. 29, 2010 10 / 22
Identifying Fixed Points Using the revised equation, we can identify three fixed points, by setting right hand side to = 0 and solving for p dp h /dt = p h (1 p h )(1 2p h ) = 0 p = 0 p = 1 p = 1/2 DR (Institute) Oct. 29, 2010 11 / 22
Assessing Stability of Fixed Points Reasoning about the revised equation, we can determine whether the three fixed points are stable: dp h /dt = p h (1 p h )(1 2p h ) = 0 p = 0 p = 1 p = 1/2 For p h and (1 p h ), all three fixed points will be positive, and will be between 0 and 1. The key term then is (1 2p h ), which will be increasing when p < 1/2 and decreasing when p > 1/2. DR (Institute) Oct. 29, 2010 12 / 22
Drawing Phase Diagrams The Hawk-Dove example is illustrated in the bi-stable phase diagram below. Other types of phase diagrams for two-strategy systems are included as well. [H]0 ->X< 0[D] Bistable (a>c, b<d) X< 0 >X Coexist (a<c b>d) 0 >X D dom H (a<c b<d) 0< X H dom D (a>c b>d) DR (Institute) Oct. 29, 2010 13 / 22
Stability Terms Asymptotically stable: starting from some initial condition, x 0, within a local neighborhood, trajectory approaches (converges back to) x. Globally stable: starting from any x 0 anywhere the system is defined, trajectory approaches (converges back to) x. Neutrally stable: starting from some x 0 within a local neighborhood, system does not approach fixed point, but does not leave local neighborhood. Unstable: not asymptotically, globally or neutrally stable, trajectory of x(t) escapes local neighborhood and stays out. DR (Institute) Oct. 29, 2010 14 / 22
Relationship of Asymptotically Stable Point to Nash Fixed point Nash Asymptotically Stable Strict Nash DR (Institute) Oct. 29, 2010 15 / 22
Applications in Law: Goods Games Segregation Ordinances and Public Cities like Baltimore and Louisville passed segregation ordinances to requires blacks and whites to live apart. What does law add to informal punishment? Public goods game where participants invest, earn a rate of return, and then divide proceeds among participants. Dynamic change: Replicator Dynamics Four strategies: x-cooperator: Invests and earns profit, does not punish y-defector: Earns profit, does not invest or punish z-loner: Doesn t participate in game, earns small fixed income w-punisher: Invests and earns profit, punishes defectors DR (Institute) Oct. 29, 2010 16 / 22
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Police and Drug Gangs: Predator-Prey Relationships Prey (rabbits) differential equation: Prey reproduce at a(x)(t), are eaten by predators at b(x(t))(y)(t) dx/dt = x(t)((a by(t)) a is rabbit birth rate constant, b is rabbit constant Predator (foxes) differential equation: Predators die at cy(t) and reproduce at dx(t)(y)(t). dy/dt = y(t)( c + dx(t)) c is fox death rate constant, d is fox birth rate constant DR (Institute) Oct. 29, 2010 18 / 22
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Comparison of EGT to Neo-classical Theorem 1: 1. If p is a Nash equilibrium of the evolutionary game, then p is a fixed point of the replicator dynamic. 2. However, if p is a fixed point of the replicator dynamic, then p need not be a Nash equilibrium of the evolutionary game. Proof: 1. The Nash equilibrium has the property that any strategy played with non-zero probability receives the same payoff. The difference term in Eq. 1, p i (π i φ), must thus be equal to zero, either because the strategy is played with probability zero or because the strategy receives the same payoff as all the others and hence the difference term will be equal to zero. 2. A state p can be a fixed point and yet not be a Nash equilibrium, when the superior best replies to p are played by zero percent of the population. In our previous example, all Hawks was a fixed point of the replicator dynamic, even though this state was not a Nash equilibrium. DR (Institute) Oct. 29, 2010 20 / 22
Comparison of EGT to Neo-classical Theorem 2: 1. If p is an evolutionary equilibrium (i.e. an asymptotically stable fixed point) of the replicator dynamics, then p is a Nash equilibrium. 2. However, if p is a Nash equilibrium, it does not need to be an evolutionary equilibrium (i.e. a stable fixed point) of the replicator dynamics. Proof: If a better reply exists to p, then perturbations from a strategy that includes these replies will yield dynamics that lead away from the fixed point. Because this is not the case, p is a Nash equilibrium. Consider a game with a Nash equilibrium in dominated strategies: X Y X...Y 1, 1 0, 0 0, 2 0, 0 The strategy (y,y) is a Nash equilibrium, but any trajectory starting at a population distribution where the proportion playing x is non-zero, will converge to the state where all players play x, and the system will never converge to y. DR (Institute) Oct. 29, 2010 21 / 22
Folk Theorems on EGT and Nash Strict Nash is always asymptotically stable. Strict Nash is always ESS. ESS is always asymptotically stable point. An asymptotically stable fixed point is always Nash. Nash is always a fixed point The converse of these theorems is false. DR (Institute) Oct. 29, 2010 22 / 22