1. 1 UNIVERSITY OF CALIFORNIA, IRVINE, GAME THEORY AND POLITICS 2, POL SCI 130B, 67130 Lecture 2.5: AGGRESSION, VIOLENCE, AND UNCERTAINTY Recall various interpretations of mixed strategies: 1. Randomization or pseudorandomization 2. Expectations/beliefs 3. Frequencies 4. Incomplete information

2. 2 Let’s assume now that we are in a Hobbesian world (think prison) where everybody fights with everybody else. When two players meet, they consider how aggressive they should be towards the other guy. Players: they may be “weak” (with probability 1-p) or “tough” (p). Strategies: Aggressive or Nice Payoffs (general rules): 1.The prize equal to 4 goes to the winner; 2.NN: Tie, i.e., the players share the prize fifty-fifty; 3.AN or NA: A wins over N; 4.AA: the cost of fight is 1 to everybody and: tough ties with tough weak ties with weak tough beats weak Q1: What happens in every specific game? Q2: What is the average payoff in the population when players know their types? Q3: What is such average payoff when players do not know their types? Violence under incomplete information

3. 3 Weak (1-p) Tough (p) 1, 14, 03, -14, 0 0, 42, 20, 42, 2 -1, 34, 01, 14, 0 0, 42, 20, 42, 2 Weak (1-p)Tough (p) A A N N A N

4. 4 Q2: When players know each other’s type, we have four different equilibria played with various probabilities and the average payoff is the sum of equilibrium payoffs multiplied by the probability that a given equilibrium occurs: P 1 = p 2 + 4p(1-p) + 0 + (1-p) 2 = - 2p 2 + 2p + 1 = 1 + 2p(1-p) Q3: When players do not know each other’s type, then we have a game of incomplete information. We will consider the case when players know nothing about their own or their opponent’s type. Then, both players may be (independently) tough with probability p and weak with 1-p. The payoffs in such a game are equal to: x, x4, 0 0, 42, 2 where x = p 2 + 3p(1-p) - p(1-p) + (1-p) 2 = 1! A N A N

5. 5 With full information, violence would be MUCH reduced! Think about: wars crime cheating during exams … Our game became __________ and the average payoff is 1 < P 1 for p ∈ (0,1) The value P 1 – 1 = 2 p(1-p) is a pure social waste due to violence resulting from the lack of information. 1, 14, 0 0, 42, 2 A N A N

6. 6 How informationally mature environments increase the quality of information and reduce violence? The case of Polish prison. 1. Self-sorting into casts (recall grypsmen, suckers, and fags) GRYPSMAN FIGHT SQ = status quo; CH = challenge; F = fight; S = surrender

7. 7 EXPLOITATION OF THE SUCKER FAG’S CARROT AND STICK SQ = status quo; CH = challenge; D = defend yourself or your property; A = accept violence or theft. R = refuse sexual service; V = volunteer; B = beat/punish; L = leave alone.

8. 8 2. Signaling your toughness and fight substitutes Fake chicken and organizing tests Argot duels and argot competence Affairs Entry Self-injuries and faking One-sided beatings versus fights When fights really happen? GENERIC SIGNALING GAME C = capable of sending a signal; I = incapable of sending a signal; S = send; N = do not send.

9. 9 Mixed strategies as proportions of species in an ecosystem Hawk-Dove games Two animals fight for a prize. They may be aggressive (Hawk) or nice (Dove). The value of the prize is 2V > 0. The cost of fighting is 2C > 0. V-C, V-C2V, 0 0, 2VV, V Animal 1 Hawk Dove Animal 2 Hawk Dove

10. 10 If V > C, then Hawk is a dominant strategy. If V < C, then there is no dominant strategy. Let’s assume that V = 1 and C = 2. -1, -12, 0 0, 21, 1 Animal 1 Hawk Dove Animal 2 Hawk Dove For V = 1 and C = 2, the game is ___________ The NE in mixed strategies is ____________

11. 11 Now, let’s change our interpretation of strategies: A strategy is not a matter of choice but it represents a specific “personality” of a player, i.e., his type of behavior or “phenotypes.” A player has no control over his personality. A “game” represents a specific “ecology” with a large number of players with different personalities. The proportions of players of different personalities are mixed strategies. Thus, p = q. We assume that the number of players is so big that adding one more player does not change the proportions. Back to our game: The payoff of a Hawk is: P H (p,p) = (-1)p + 2(1-p) = 2 – 3p The payoff of a Dove is: P D (p,p) = 1 – p A player can “invade” a population of other players if she gets a higher expected payoff against them than players in the original population. Consider a population of Hawks: Can a Dove “invade”? Consider a population of Doves: Can a Hawk “invade”? Consider a population of ½ Hawks and ½ Doves Can a single Hawk or Dove invade? Can any cluster of Hawks or Doves invade?

12. 12 What happens when the population of Hawks or Doves falls below 1/2 ?

13. 13 CAVEAT: Sensitivity of equilibria to payoffs : Let’s check how the equilibrium depends on our payoff assumptions: The game is still Chicken if and only if x _______________ In such a case, NE #1 and NE #2 remain unchanged NE #3 (in mixed strategies) ______________________ x, x2, 0 0, 21, 1 Animal 1 Hawk Dove Animal 2 Hawk Dove

14. 14 Summary: We can model an encounter of two potentially aggressive players as a game of incomplete information. When there is uncertainty about the level of players’ aggressiveness or toughness, inefficient outcomes, such as violent fights, may arise. The legal system, social norms as well as the informal norms of various subcultures evolve in order to minimize the costs of violence. In ecosystems, mixed strategies may denote the proportions of different types of behavior. An interesting fact is that the NE in mixed strategies in Chicken is an “attractor,” i.e., the proportions of different types of behavior tend to stabilize at the NE level.