Cooperation in Con‡ict Games Sandeep Baliga and Tomas Sjöström MEDS and Rutgers December 2007 Baliga and Sjöström (MEDS and Rutgers) Cooperation in Con‡ict Games 12/07 1 / 17 Stag Hunt: aggression feeds on itself and escalates con‡ict (actions are strategic complements). Chicken: toughness deters aggression (actions are strategic substitutes) “World War I was an unwanted spiral of hostility”...“World War II was not an unwanted spiral of hostility-it was a failure to deter Hitler’splanned aggression.”(Joseph Nye (2007)). Stag hunt and chicken have multiple Nash equilibria. Jervis (spiral model), Schelling (reciprocal fear of surprise attack): mutual fear and uncertainty determine the outcome. Introduction In international relations, Stag Hunt, Chicken and Prisoners’Dilemma represent canonical strategic interactions. Baliga and Sjöström (MEDS and Rutgers) Cooperation in Con‡ict Games 12/07 2 / 17 Chicken: toughness deters aggression (actions are strategic substitutes) “World War I was an unwanted spiral of hostility”...“World War II was not an unwanted spiral of hostility-it was a failure to deter Hitler’splanned aggression.”(Joseph Nye (2007)). Stag hunt and chicken have multiple Nash equilibria. Jervis (spiral model), Schelling (reciprocal fear of surprise attack): mutual fear and uncertainty determine the outcome. Introduction In international relations, Stag Hunt, Chicken and Prisoners’Dilemma represent canonical strategic interactions. Stag Hunt: aggression feeds on itself and escalates con‡ict (actions are strategic complements). Baliga and Sjöström (MEDS and Rutgers) Cooperation in Con‡ict Games 12/07 2 / 17 “World War I was an unwanted spiral of hostility”...“World War II was not an unwanted spiral of hostility-it was a failure to deter Hitler’splanned aggression.”(Joseph Nye (2007)). Stag hunt and chicken have multiple Nash equilibria. Jervis (spiral model), Schelling (reciprocal fear of surprise attack): mutual fear and uncertainty determine the outcome. Introduction In international relations, Stag Hunt, Chicken and Prisoners’Dilemma represent canonical strategic interactions. Stag Hunt: aggression feeds on itself and escalates con‡ict (actions are strategic complements). Chicken: toughness deters aggression (actions are strategic substitutes) Baliga and Sjöström (MEDS and Rutgers) Cooperation in Con‡ict Games 12/07 2 / 17 Stag hunt and chicken have multiple Nash equilibria. Jervis (spiral model), Schelling (reciprocal fear of surprise attack): mutual fear and uncertainty determine the outcome. Introduction In international relations, Stag Hunt, Chicken and Prisoners’Dilemma represent canonical strategic interactions. Stag Hunt: aggression feeds on itself and escalates con‡ict (actions are strategic complements). Chicken: toughness deters aggression (actions are strategic substitutes) “World War I was an unwanted spiral of hostility”...“World War II was not an unwanted spiral of hostility-it was a failure to deter Hitler’splanned aggression.”(Joseph Nye (2007)). Baliga and Sjöström (MEDS and Rutgers) Cooperation in Con‡ict Games 12/07 2 / 17 Introduction In international relations, Stag Hunt, Chicken and Prisoners’Dilemma represent canonical strategic interactions. Stag Hunt: aggression feeds on itself and escalates con‡ict (actions are strategic complements). Chicken: toughness deters aggression (actions are strategic substitutes) “World War I was an unwanted spiral of hostility”...“World War II was not an unwanted spiral of hostility-it was a failure to deter Hitler’splanned aggression.”(Joseph Nye (2007)). Stag hunt and chicken have multiple Nash equilibria. Jervis (spiral model), Schelling (reciprocal fear of surprise attack): mutual fear and uncertainty determine the outcome. Baliga and Sjöström (MEDS and Rutgers) Cooperation in Con‡ict Games 12/07 2 / 17 Correlation: Con‡ict concerns ownership of some resource and payo¤s are correlated. Enough private information (about preferences or costs) selects a unique equilibrium. (Unlike global games, no need to assume highly correlated private information.) Are long-run relationships more peaceful? Is democracy good for peace? (Other issues: communication? mediation? signalling?) Private Information Independence: Leaders preferences for con‡ict are independent and unrelated. Baliga and Sjöström (MEDS and Rutgers) Cooperation in Con‡ict Games 12/07 3 / 17 Enough private information (about preferences or costs) selects a unique equilibrium. (Unlike global games, no need to assume highly correlated private information.) Are long-run relationships more peaceful? Is democracy good for peace? (Other issues: communication? mediation? signalling?) Private Information Independence: Leaders preferences for con‡ict are independent and unrelated. Correlation: Con‡ict concerns ownership of some resource and payo¤s are correlated. Baliga and Sjöström (MEDS and Rutgers) Cooperation in Con‡ict Games 12/07 3 / 17 Are long-run relationships more peaceful? Is democracy good for peace? (Other issues: communication? mediation? signalling?) Private Information Independence: Leaders preferences for con‡ict are independent and unrelated. Correlation: Con‡ict concerns ownership of some resource and payo¤s are correlated. Enough private information (about preferences or costs) selects a unique equilibrium. (Unlike global games, no need to assume highly correlated private information.) Baliga and Sjöström (MEDS and Rutgers) Cooperation in Con‡ict Games 12/07 3 / 17 Private Information Independence: Leaders preferences for con‡ict are independent and unrelated. Correlation: Con‡ict concerns ownership of some resource and payo¤s are correlated. Enough private information (about preferences or costs) selects a unique equilibrium. (Unlike global games, no need to assume highly correlated private information.) Are long-run relationships more peaceful? Is democracy good for peace? (Other issues: communication? mediation? signalling?) Baliga and Sjöström (MEDS and Rutgers) Cooperation in Con‡ict Games 12/07 3 / 17 The one-shot game Payo¤s for player i (row-chooser) HD H hi c hi (1) D d 0 Strategic complements (Stag Hunt): (0 <) c < d. Strategic substitutes (Chicken): c > d (> 0). We allow hi to be negative or positive (more soon). Player i’s hostility type hi is made up of a known publicly known component ki and a private idiosyncratic component η [η, η¯ ]. So, i 2 hi = ki + η . Distribution of η conditional on η = y is F ( y). i j i j Assumption 1. (i) F1(x y) > 0 (ii) F2(x y) 0 (a more hostile player i is more pessimistic aboutj j’shostility). j Notice this allows for both independent and positively correlated types. Baliga and Sjöström (MEDS and Rutgers) Cooperation in Con‡ict Games 12/07 4 / 17 Unique equilibrium (h˜i , h˜j ) is interior: h˜i + (d c) 1 F (h˜j kj h˜i ki ) = 0 j Strategic Complements Strategic complements: H more appealing if the opponent is expected to choose H (c < d) Player i is a dominant strategy hawk if hi 0. Player i is a dominant strategy dove if hi c d or hi c d. In between, we have coordination types. Theorem Suppose both players have dominant strategy hawks and doves. If d > c 1 and F1(s t) + F2(s t) < d c then the game has a unique BNE. This BNE is in cut-o¤j strategies.j Independence: “Large idiosyncratic uncertainty” implies uniqueness. This is our formalization of Schelling’s“reciprocal fear of surprise attack”. Positive correlation helps uniqueness here. Baliga and Sjöström (MEDS and Rutgers) Cooperation in Con‡ict Games 12/07 4 / 17 Strategic Complements Strategic complements: H more appealing if the opponent is expected to choose H (c < d) Player i is a dominant strategy hawk if hi 0. Player i is a dominant strategy dove if hi c d or hi c d. In between, we have coordination types. Theorem Suppose both players have dominant strategy hawks and doves. If d > c 1 and F1(s t) + F2(s t) < d c then the game has a unique BNE. This BNE is in cut-o¤j strategies.j Independence: “Large idiosyncratic uncertainty” implies uniqueness. This is our formalization of Schelling’s“reciprocal fear of surprise attack”. Positive correlation helps uniqueness here. Unique equilibrium (h˜i , h˜j ) is interior: h˜i + (d c) 1 F (h˜j kj h˜i ki ) = 0 j Baliga and Sjöström (MEDS and Rutgers) Cooperation in Con‡ict Games 12/07 4 / 17 Player i’snet gain from choosing H instead of D when his type is hi = y is i Ψ (x, y) y + (d c) (1 F (x kj y ki )) . (2) j For a cuto¤ strategy to be a best response, player i should be more inclined to choose H the more hostile he is: i Ψ (x, y) = 1 (d c)F2(x kj y ki ) > 0 (3) 2 j In view of Assumption 1, (3) holds if d > c. (It also holds if d < c and types are not too strongly correlated.) If condition (3) holds then player i’sbest response to player j’scuto¤ x is to use a cuto¤ point denoted βi (x). If player j uses a “cuto¤ strategy” with cuto¤ hostility type hj = x, then for player i with hostility type hi = y, the probability player j plays D is δj (y) = F (x kj y ki ) j Baliga and Sjöström (MEDS and Rutgers) Cooperation in Con‡ict Games 12/07 5 / 17 For a cuto¤ strategy to be a best response, player i should be more inclined to choose H the more hostile he is: i Ψ (x, y) = 1 (d c)F2(x kj y ki ) > 0 (3) 2 j In view of Assumption 1, (3) holds if d > c. (It also holds if d < c and types are not too strongly correlated.) If condition (3) holds then player i’sbest response to player j’scuto¤ x is to use a cuto¤ point denoted βi (x). If player j uses a “cuto¤ strategy” with cuto¤ hostility type hj = x, then for player i with hostility type hi = y, the probability player j plays D is δj (y) = F (x kj y ki ) j Player i’snet gain from choosing H instead of D when his type is hi = y is i Ψ (x, y) y + (d c) (1 F (x kj y ki )) .