Examples of Equilibrium Strategies (2) 3. Equilibrium that depends on cardinal preferences: Assume two rounds of communication. Round 1: Each agent sends moderate or extreme . Uniform distribution: moderate iff u ∈ ( 0 . 25 , 0 . 75 ) . Round 2 and actions: Two extremists: babble; each agent plays his preferred direction. Extremist & moderate: coordinate on the extremist’s preferred outcome. Two moderates: report preferred directions; if disagree, coordinated outcome is chosen by a joint lottery. Heller & Kuzmics Renegotiation & Coordination March 2020 13 / 33
Examples of Equilibrium Strategies (2) 3. Equilibrium that depends on cardinal preferences: Assume two rounds of communication. Round 1: Each agent sends moderate or extreme . Uniform distribution: moderate iff u ∈ ( 0 . 25 , 0 . 75 ) . Round 2 and actions: Two extremists: babble; each agent plays his preferred direction. Extremist & moderate: coordinate on the extremist’s preferred outcome. Two moderates: report preferred directions; if disagree, coordinated outcome is chosen by a joint lottery. Heller & Kuzmics Renegotiation & Coordination March 2020 13 / 33
Three Properties of Equilibria (Satisfied by σ L , σ R , σ C ) 1 Coordinated : Players never mis-coordinate (i.e, never play ( L , R ) ). 2 Mutual-preference consistent : If both agents prefer the same outcome, they always play it. 3 (Essentially) binary communication: The message of any type u < 0 . 5 has the same impact: maximizing the probability that the opponent plays L . The message of any type u > 0 . 5 has the same impact: minimizing the probability that the opponent plays L . Formal Def. Heller & Kuzmics Renegotiation & Coordination March 2020 14 / 33
Three Properties of Equilibria (Satisfied by σ L , σ R , σ C ) 1 Coordinated : Players never mis-coordinate (i.e, never play ( L , R ) ). 2 Mutual-preference consistent : If both agents prefer the same outcome, they always play it. 3 (Essentially) binary communication: The message of any type u < 0 . 5 has the same impact: maximizing the probability that the opponent plays L . The message of any type u > 0 . 5 has the same impact: minimizing the probability that the opponent plays L . Formal Def. Heller & Kuzmics Renegotiation & Coordination March 2020 14 / 33
Three Properties of Equilibria (Satisfied by σ L , σ R , σ C ) 1 Coordinated : Players never mis-coordinate (i.e, never play ( L , R ) ). 2 Mutual-preference consistent : If both agents prefer the same outcome, they always play it. 3 (Essentially) binary communication: The message of any type u < 0 . 5 has the same impact: maximizing the probability that the opponent plays L . The message of any type u > 0 . 5 has the same impact: minimizing the probability that the opponent plays L . Formal Def. Heller & Kuzmics Renegotiation & Coordination March 2020 14 / 33
1-Dimensional Set of Strategies Satisfying the 3 Properties Definition The left-tendency of a strategy: the probability of coordination on L conditional on the players having different preferred outcomes. Any strategy satisfying the above three properties is characterized by its left-tendency α (e.g., α = 1 ↔ σ L , α = 0 ↔ σ R , α = 0 . 5 ↔ σ C ): Two types u , v < 0 . 5: play L . Two types u , v > 0 . 5: play R . Two opposing types u < 0 . 5 < v : joint lottery, coordinate on L with probability α , and coordinate on R with probability 1 − α . Heller & Kuzmics Renegotiation & Coordination March 2020 15 / 33
1-Dimensional Set of Strategies Satisfying the 3 Properties Definition The left-tendency of a strategy: the probability of coordination on L conditional on the players having different preferred outcomes. Any strategy satisfying the above three properties is characterized by its left-tendency α (e.g., α = 1 ↔ σ L , α = 0 ↔ σ R , α = 0 . 5 ↔ σ C ): Two types u , v < 0 . 5: play L . Two types u , v > 0 . 5: play R . Two opposing types u < 0 . 5 < v : joint lottery, coordinate on L with probability α , and coordinate on R with probability 1 − α . Heller & Kuzmics Renegotiation & Coordination March 2020 15 / 33
1-Dimensional Set of Strategies Satisfying the 3 Properties Definition The left-tendency of a strategy: the probability of coordination on L conditional on the players having different preferred outcomes. Any strategy satisfying the above three properties is characterized by its left-tendency α (e.g., α = 1 ↔ σ L , α = 0 ↔ σ R , α = 0 . 5 ↔ σ C ): Two types u , v < 0 . 5: play L . Two types u , v > 0 . 5: play R . Two opposing types u < 0 . 5 < v : joint lottery, coordinate on L with probability α , and coordinate on R with probability 1 − α . Heller & Kuzmics Renegotiation & Coordination March 2020 15 / 33
Renegotiation-Proofness (Informal Definition) Players may further communicate & renegotiate to a Pareto-better equilibrium after observing the pair of messages sent in the first round. Renegotiation-proofness: There is no other equilibrium that Pareto-dominates the existing equilibrium. Motivation: Otherwise, agents renegotiate to the better equilibrium o m & Myerson, 1983; Maskin & Farrell, 1989; Benoit & Krishna, 1993 ). (Holmstr ¨ Evolutionary motivation: otherwise, self-enforcing secret handshake , (Hamilton, 1964; Dawkins, 1976; Robson, 1990; equilibrium entrants - Swinkels, 1992; collaboration - Newton, 2017) . Heller & Kuzmics Renegotiation & Coordination March 2020 16 / 33
Renegotiation-Proofness (Informal Definition) Players may further communicate & renegotiate to a Pareto-better equilibrium after observing the pair of messages sent in the first round. Renegotiation-proofness: There is no other equilibrium that Pareto-dominates the existing equilibrium. Motivation: Otherwise, agents renegotiate to the better equilibrium o m & Myerson, 1983; Maskin & Farrell, 1989; Benoit & Krishna, 1993 ). (Holmstr ¨ Evolutionary motivation: otherwise, self-enforcing secret handshake , (Hamilton, 1964; Dawkins, 1976; Robson, 1990; equilibrium entrants - Swinkels, 1992; collaboration - Newton, 2017) . Heller & Kuzmics Renegotiation & Coordination March 2020 16 / 33
Renegotiation-Proofness (Informal Definition) Players may further communicate & renegotiate to a Pareto-better equilibrium after observing the pair of messages sent in the first round. Renegotiation-proofness: There is no other equilibrium that Pareto-dominates the existing equilibrium. Motivation: Otherwise, agents renegotiate to the better equilibrium o m & Myerson, 1983; Maskin & Farrell, 1989; Benoit & Krishna, 1993 ). (Holmstr ¨ Evolutionary motivation: otherwise, self-enforcing secret handshake , (Hamilton, 1964; Dawkins, 1976; Robson, 1990; equilibrium entrants - Swinkels, 1992; collaboration - Newton, 2017) . Heller & Kuzmics Renegotiation & Coordination March 2020 16 / 33
Induced Games with Additional Communication Assume that both players send 1st-stage messages according to µ . Each message m ∈ supp ( µ ) induces a posterior probability F m for the player’s type, conditional on the player sending message m . Each pair m , m ′ ∈ supp ( µ ) induces a coordination game without communication Γ( F m , F m ′ ) in which the players’ types are distributed according to F m and F m ′ . Let Γ( F m , F m ′ , M ) be the induced game that is modified to allow the players to have an additional round of communication (called, induced game with additional communication). Heller & Kuzmics Renegotiation & Coordination March 2020 17 / 33
Induced Games with Additional Communication Assume that both players send 1st-stage messages according to µ . Each message m ∈ supp ( µ ) induces a posterior probability F m for the player’s type, conditional on the player sending message m . Each pair m , m ′ ∈ supp ( µ ) induces a coordination game without communication Γ( F m , F m ′ ) in which the players’ types are distributed according to F m and F m ′ . Let Γ( F m , F m ′ , M ) be the induced game that is modified to allow the players to have an additional round of communication (called, induced game with additional communication). Heller & Kuzmics Renegotiation & Coordination March 2020 17 / 33
Induced Games with Additional Communication Assume that both players send 1st-stage messages according to µ . Each message m ∈ supp ( µ ) induces a posterior probability F m for the player’s type, conditional on the player sending message m . Each pair m , m ′ ∈ supp ( µ ) induces a coordination game without communication Γ( F m , F m ′ ) in which the players’ types are distributed according to F m and F m ′ . Let Γ( F m , F m ′ , M ) be the induced game that is modified to allow the players to have an additional round of communication (called, induced game with additional communication). Heller & Kuzmics Renegotiation & Coordination March 2020 17 / 33
Renegotiation-Proofness (Formal Definition) Definition Equilibrium ( σ m , σ m ′ ) of Γ( F m , F m ′ , M ) Pareto-dominates equilibrium ( x , x ′ ) of Γ( F m , F m ′ ) if the former induces a weakly higher payoff than the latter for each type in the support of F m / F m ′ of each player, with a strict inequality for some types. Definition Equilibrium strategy σ = ( µ , ξ ) is renegotiation-proof (RP) if for each pair of messages m , m ′ ∈ supp ( µ ) , the equilibrium ( ξ ( m , m ′ ) , ξ ( m ′ , m )) of Γ( F m , F m ′ ) is not Pareto-dominated by any equilibrium of Γ( F m , F m ′ , M ) . Heller & Kuzmics Renegotiation & Coordination March 2020 18 / 33
Renegotiation-Proofness (Formal Definition) Definition Equilibrium ( σ m , σ m ′ ) of Γ( F m , F m ′ , M ) Pareto-dominates equilibrium ( x , x ′ ) of Γ( F m , F m ′ ) if the former induces a weakly higher payoff than the latter for each type in the support of F m / F m ′ of each player, with a strict inequality for some types. Definition Equilibrium strategy σ = ( µ , ξ ) is renegotiation-proof (RP) if for each pair of messages m , m ′ ∈ supp ( µ ) , the equilibrium ( ξ ( m , m ′ ) , ξ ( m ′ , m )) of Γ( F m , F m ′ ) is not Pareto-dominated by any equilibrium of Γ( F m , F m ′ , M ) . Heller & Kuzmics Renegotiation & Coordination March 2020 18 / 33
Outline Introduction 1 Model 2 Main result 3 Efficiency and stability 4 Extensions & discussion 5
Main Result Theorem Strategy σ ∗ is renegotiation-proof equilibrium strategy iff it is 1 coordinated, 2 mutual-preference consistent, 3 binary communication. Heller & Kuzmics Renegotiation & Coordination March 2020 20 / 33
Intuition for the Main Result (RP ⇒ 3 Key Properties) 1 Coordinated : Any miscoordinated equilibrium of an induced coordination game Γ( F m , F m ′ ) can be Pareto-improved by either σ L , σ R , or σ C . Proof 2 Mutual-preference consistent : If not, it can be Pareto-improved by σ L / σ R . 3 Binary communication : Coordinated ⇒ agent cares only for the average partner’s probability of playing L . Agent with u ≤ 0 . 5 sends m ∈ M L ⇒ the message m maximizes this probability. Any m , m ′ ∈ M L induce the same probability of a partner who prefers R to play L . Heller & Kuzmics Renegotiation & Coordination March 2020 21 / 33
Intuition for the Main Result (RP ⇒ 3 Key Properties) 1 Coordinated : Any miscoordinated equilibrium of an induced coordination game Γ( F m , F m ′ ) can be Pareto-improved by either σ L , σ R , or σ C . Proof 2 Mutual-preference consistent : If not, it can be Pareto-improved by σ L / σ R . 3 Binary communication : Coordinated ⇒ agent cares only for the average partner’s probability of playing L . Agent with u ≤ 0 . 5 sends m ∈ M L ⇒ the message m maximizes this probability. Any m , m ′ ∈ M L induce the same probability of a partner who prefers R to play L . Heller & Kuzmics Renegotiation & Coordination March 2020 21 / 33
Intuition for the Main Result (RP ⇒ 3 Key Properties) 1 Coordinated : Any miscoordinated equilibrium of an induced coordination game Γ( F m , F m ′ ) can be Pareto-improved by either σ L , σ R , or σ C . Proof 2 Mutual-preference consistent : If not, it can be Pareto-improved by σ L / σ R . 3 Binary communication : Coordinated ⇒ agent cares only for the average partner’s probability of playing L . Agent with u ≤ 0 . 5 sends m ∈ M L ⇒ the message m maximizes this probability. Any m , m ′ ∈ M L induce the same probability of a partner who prefers R to play L . Heller & Kuzmics Renegotiation & Coordination March 2020 21 / 33
Intuition for the Main Result (RP ⇒ 3 Key Properties) 1 Coordinated : Any miscoordinated equilibrium of an induced coordination game Γ( F m , F m ′ ) can be Pareto-improved by either σ L , σ R , or σ C . Proof 2 Mutual-preference consistent : If not, it can be Pareto-improved by σ L / σ R . 3 Binary communication : Coordinated ⇒ agent cares only for the average partner’s probability of playing L . Agent with u ≤ 0 . 5 sends m ∈ M L ⇒ the message m maximizes this probability. Any m , m ′ ∈ M L induce the same probability of a partner who prefers R to play L . Heller & Kuzmics Renegotiation & Coordination March 2020 21 / 33
Intuition for the Main Result (RP ⇒ 3 Key Properties) 1 Coordinated : Any miscoordinated equilibrium of an induced coordination game Γ( F m , F m ′ ) can be Pareto-improved by either σ L , σ R , or σ C . Proof 2 Mutual-preference consistent : If not, it can be Pareto-improved by σ L / σ R . 3 Binary communication : Coordinated ⇒ agent cares only for the average partner’s probability of playing L . Agent with u ≤ 0 . 5 sends m ∈ M L ⇒ the message m maximizes this probability. Any m , m ′ ∈ M L induce the same probability of a partner who prefers R to play L . Heller & Kuzmics Renegotiation & Coordination March 2020 21 / 33
Intuition for the Main Result (3 Key Properties ⇒ RP ) Showing that σ = ( µ , ξ ) satisfying the 3 properties is an equilibrium strategy: Coordinated ⇒ Best reply in stage 2 = matching the partner’s pure action = follow ξ . Binary communication ⇒ Best reply in stage 1 = maximizing the probability to coordinate on the preferred outcome = follow µ . Showing that σ = ( µ , ξ ) satisfies renegotiation-proofness: After communicating, at least one of the players gets his maximal feasible payoff ⇒ The equilibrium of the induced game is not Pareto-dominated. Heller & Kuzmics Renegotiation & Coordination March 2020 22 / 33
Intuition for the Main Result (3 Key Properties ⇒ RP ) Showing that σ = ( µ , ξ ) satisfying the 3 properties is an equilibrium strategy: Coordinated ⇒ Best reply in stage 2 = matching the partner’s pure action = follow ξ . Binary communication ⇒ Best reply in stage 1 = maximizing the probability to coordinate on the preferred outcome = follow µ . Showing that σ = ( µ , ξ ) satisfies renegotiation-proofness: After communicating, at least one of the players gets his maximal feasible payoff ⇒ The equilibrium of the induced game is not Pareto-dominated. Heller & Kuzmics Renegotiation & Coordination March 2020 22 / 33
Intuition for the Main Result (3 Key Properties ⇒ RP ) Showing that σ = ( µ , ξ ) satisfying the 3 properties is an equilibrium strategy: Coordinated ⇒ Best reply in stage 2 = matching the partner’s pure action = follow ξ . Binary communication ⇒ Best reply in stage 1 = maximizing the probability to coordinate on the preferred outcome = follow µ . Showing that σ = ( µ , ξ ) satisfies renegotiation-proofness: After communicating, at least one of the players gets his maximal feasible payoff ⇒ The equilibrium of the induced game is not Pareto-dominated. Heller & Kuzmics Renegotiation & Coordination March 2020 22 / 33
Outline Introduction 1 Model 2 Main result 3 Efficiency and stability 4 Extensions & discussion 5
On Ex-Ante Optimality of σ L / σ R First-best (play L iff u + v < 1) is not an equilibrium behavior. Each agent would claim to have an extreme type. Non-RP Equilibria with mis-coordination allow the outcome to depend on the intensity of preferences. Given some distributions of types, this may yield a higher ex-ante payoff. Example Proposition Either σ L or σ R : Improves ex-ante payoff relative to all no-communication equilibria. Maximizes ex-ante payoff among all coordinated equilibria. Heller & Kuzmics Renegotiation & Coordination March 2020 24 / 33
On Ex-Ante Optimality of σ L / σ R First-best (play L iff u + v < 1) is not an equilibrium behavior. Each agent would claim to have an extreme type. Non-RP Equilibria with mis-coordination allow the outcome to depend on the intensity of preferences. Given some distributions of types, this may yield a higher ex-ante payoff. Example Proposition Either σ L or σ R : Improves ex-ante payoff relative to all no-communication equilibria. Maximizes ex-ante payoff among all coordinated equilibria. Heller & Kuzmics Renegotiation & Coordination March 2020 24 / 33
On Ex-Ante Optimality of σ L / σ R First-best (play L iff u + v < 1) is not an equilibrium behavior. Each agent would claim to have an extreme type. Non-RP Equilibria with mis-coordination allow the outcome to depend on the intensity of preferences. Given some distributions of types, this may yield a higher ex-ante payoff. Example Proposition Either σ L or σ R : Improves ex-ante payoff relative to all no-communication equilibria. Maximizes ex-ante payoff among all coordinated equilibria. Heller & Kuzmics Renegotiation & Coordination March 2020 24 / 33
Interim Pareto-Optimality of RP Equilibrium Strategies The definition of RP requires a mild notion of Pareto efficiency: post-communication Pareto efficiency WRT equilibrium strategies. Proposition Any RP equilibrium strategy satisfies interim (pre/post-communication) Pareto efficiency WRT all strategy profiles. Intuition: Any payoff increase to a positive measure of “left” types must decrease the payoff of a positive measure of “right” types. Heller & Kuzmics Renegotiation & Coordination March 2020 25 / 33
Interim Pareto-Optimality of RP Equilibrium Strategies The definition of RP requires a mild notion of Pareto efficiency: post-communication Pareto efficiency WRT equilibrium strategies. Proposition Any RP equilibrium strategy satisfies interim (pre/post-communication) Pareto efficiency WRT all strategy profiles. Intuition: Any payoff increase to a positive measure of “left” types must decrease the payoff of a positive measure of “right” types. Heller & Kuzmics Renegotiation & Coordination March 2020 25 / 33
Ex-ante & Pre-Communication Renegotiation-Proofness Our definition of RP is “post-communication”: agents are allowed to renegotiate only after the original communication. Alternative definitions: Pre-communication RP: agents can renegotiate also before communicating. Ex-ante RP: agents can renegot. also before knowing their own types. Proposition The set of pre-communication RP = the set of post-communication RP. Either σ L or σ R is the unique ex-ante RP equilibrium strategy (unless both σ L and σ R induce the same ex-ante payoff) .
Ex-ante & Pre-Communication Renegotiation-Proofness Our definition of RP is “post-communication”: agents are allowed to renegotiate only after the original communication. Alternative definitions: Pre-communication RP: agents can renegotiate also before communicating. Ex-ante RP: agents can renegot. also before knowing their own types. Proposition The set of pre-communication RP = the set of post-communication RP. Either σ L or σ R is the unique ex-ante RP equilibrium strategy (unless both σ L and σ R induce the same ex-ante payoff) .
Ex-ante & Pre-Communication Renegotiation-Proofness Our definition of RP is “post-communication”: agents are allowed to renegotiate only after the original communication. Alternative definitions: Pre-communication RP: agents can renegotiate also before communicating. Ex-ante RP: agents can renegot. also before knowing their own types. Proposition The set of pre-communication RP = the set of post-communication RP. Either σ L or σ R is the unique ex-ante RP equilibrium strategy (unless both σ L and σ R induce the same ex-ante payoff) .
Ex-ante & Pre-Communication Renegotiation-Proofness Our definition of RP is “post-communication”: agents are allowed to renegotiate only after the original communication. Alternative definitions: Pre-communication RP: agents can renegotiate also before communicating. Ex-ante RP: agents can renegot. also before knowing their own types. Proposition The set of pre-communication RP = the set of post-communication RP. Either σ L or σ R is the unique ex-ante RP equilibrium strategy (unless both σ L and σ R induce the same ex-ante payoff) .
Ex-ante & Pre-Communication Renegotiation-Proofness Our definition of RP is “post-communication”: agents are allowed to renegotiate only after the original communication. Alternative definitions: Pre-communication RP: agents can renegotiate also before communicating. Ex-ante RP: agents can renegot. also before knowing their own types. Proposition The set of pre-communication RP = the set of post-communication RP. Either σ L or σ R is the unique ex-ante RP equilibrium strategy (unless both σ L and σ R induce the same ex-ante payoff) .
Any RP Eq. Strategy ( µ , ξ ) is Robust to Perturbations: 1 Neutral stability (Maynard smith & Price, 1973) ⇒ Robustness to a few experimenting agents. Details µ is weakly dominant, given the second-stage behavior ξ ⇒ Robustness to 2 any perturbation that changes the 1st-stage behavior. Details 3 ξ is a neighborhood invader strategy in each induced game (Cressman, 2010; a refinement of CSS ` a la Eshel & Motro, 1981) ⇒ Robustness to any sufficiently small perturbation in the 2nd-stage behavior. Details Illustration Some non-RP strategies may satisfy (1–3) (e.g., the uniform norm always- L satisfies (1+2), and for some distributions of types it satisfies (3). Heller & Kuzmics Renegotiation & Coordination March 2020 27 / 33
Any RP Eq. Strategy ( µ , ξ ) is Robust to Perturbations: 1 Neutral stability (Maynard smith & Price, 1973) ⇒ Robustness to a few experimenting agents. Details µ is weakly dominant, given the second-stage behavior ξ ⇒ Robustness to 2 any perturbation that changes the 1st-stage behavior. Details 3 ξ is a neighborhood invader strategy in each induced game (Cressman, 2010; a refinement of CSS ` a la Eshel & Motro, 1981) ⇒ Robustness to any sufficiently small perturbation in the 2nd-stage behavior. Details Illustration Some non-RP strategies may satisfy (1–3) (e.g., the uniform norm always- L satisfies (1+2), and for some distributions of types it satisfies (3). Heller & Kuzmics Renegotiation & Coordination March 2020 27 / 33
Any RP Eq. Strategy ( µ , ξ ) is Robust to Perturbations: 1 Neutral stability (Maynard smith & Price, 1973) ⇒ Robustness to a few experimenting agents. Details µ is weakly dominant, given the second-stage behavior ξ ⇒ Robustness to 2 any perturbation that changes the 1st-stage behavior. Details 3 ξ is a neighborhood invader strategy in each induced game (Cressman, 2010; a refinement of CSS ` a la Eshel & Motro, 1981) ⇒ Robustness to any sufficiently small perturbation in the 2nd-stage behavior. Details Illustration Some non-RP strategies may satisfy (1–3) (e.g., the uniform norm always- L satisfies (1+2), and for some distributions of types it satisfies (3). Heller & Kuzmics Renegotiation & Coordination March 2020 27 / 33
Any RP Eq. Strategy ( µ , ξ ) is Robust to Perturbations: 1 Neutral stability (Maynard smith & Price, 1973) ⇒ Robustness to a few experimenting agents. Details µ is weakly dominant, given the second-stage behavior ξ ⇒ Robustness to 2 any perturbation that changes the 1st-stage behavior. Details 3 ξ is a neighborhood invader strategy in each induced game (Cressman, 2010; a refinement of CSS ` a la Eshel & Motro, 1981) ⇒ Robustness to any sufficiently small perturbation in the 2nd-stage behavior. Details Illustration Some non-RP strategies may satisfy (1–3) (e.g., the uniform norm always- L satisfies (1+2), and for some distributions of types it satisfies (3). Heller & Kuzmics Renegotiation & Coordination March 2020 27 / 33
Outline Introduction 1 Model 2 Main result 3 Efficiency and stability 4 Extensions & discussion 5
Variants and Extensions Skip to insights All results hold in the following extensions: Multiple rounds of communication. 1 n > 2 players (positive payoff iff everyone coordinates on the same action). 2 Any (possibly asymmetric) coordination game in which the payoff-dominant 3 action is risk dominant (i.e., U 11 > U 22 ⇒ 0 . 5 ( U 11 + U 12 ) > 0 . 5 ( U 21 + U 22 ) . Extreme types with dominant actions: Essentially, a unique RP eq. strategy σ α ∗ ( = σ C if the distribution of types is symmetric). Details More than two actions: (1) σ C is a RP equilibrium strategy; and (2) any RP equilibrium strategy must be: same-message coordinated and mutual preference consistent. Details Heller & Kuzmics Renegotiation & Coordination March 2020 29 / 33
Variants and Extensions Skip to insights All results hold in the following extensions: Multiple rounds of communication. 1 n > 2 players (positive payoff iff everyone coordinates on the same action). 2 Any (possibly asymmetric) coordination game in which the payoff-dominant 3 action is risk dominant (i.e., U 11 > U 22 ⇒ 0 . 5 ( U 11 + U 12 ) > 0 . 5 ( U 21 + U 22 ) . Extreme types with dominant actions: Essentially, a unique RP eq. strategy σ α ∗ ( = σ C if the distribution of types is symmetric). Details More than two actions: (1) σ C is a RP equilibrium strategy; and (2) any RP equilibrium strategy must be: same-message coordinated and mutual preference consistent. Details Heller & Kuzmics Renegotiation & Coordination March 2020 29 / 33
Variants and Extensions Skip to insights All results hold in the following extensions: Multiple rounds of communication. 1 n > 2 players (positive payoff iff everyone coordinates on the same action). 2 Any (possibly asymmetric) coordination game in which the payoff-dominant 3 action is risk dominant (i.e., U 11 > U 22 ⇒ 0 . 5 ( U 11 + U 12 ) > 0 . 5 ( U 21 + U 22 ) . Extreme types with dominant actions: Essentially, a unique RP eq. strategy σ α ∗ ( = σ C if the distribution of types is symmetric). Details More than two actions: (1) σ C is a RP equilibrium strategy; and (2) any RP equilibrium strategy must be: same-message coordinated and mutual preference consistent. Details Heller & Kuzmics Renegotiation & Coordination March 2020 29 / 33
Variants and Extensions Skip to insights All results hold in the following extensions: Multiple rounds of communication. 1 n > 2 players (positive payoff iff everyone coordinates on the same action). 2 Any (possibly asymmetric) coordination game in which the payoff-dominant 3 action is risk dominant (i.e., U 11 > U 22 ⇒ 0 . 5 ( U 11 + U 12 ) > 0 . 5 ( U 21 + U 22 ) . Extreme types with dominant actions: Essentially, a unique RP eq. strategy σ α ∗ ( = σ C if the distribution of types is symmetric). Details More than two actions: (1) σ C is a RP equilibrium strategy; and (2) any RP equilibrium strategy must be: same-message coordinated and mutual preference consistent. Details Heller & Kuzmics Renegotiation & Coordination March 2020 29 / 33
Economic Insights Little communication (1-2 bits) significantly alters the predicted play. It is easy to credibly reveal the ordinal preferences. Players cannot credibly reveal the intensity of preferences. Implications to anti-trust policy: Successful collusion often depend on the firms’ private preferences (e.g., market sharing agreement prefer to serve). Our findings strengthens the importance of not allowing even a brief form of explicit communication between oligopolistic competitors. E.g., 1997 series of FCC ascending auctions (Cramton & Schwartz, 2000). Heller & Kuzmics Renegotiation & Coordination March 2020 30 / 33
Economic Insights Little communication (1-2 bits) significantly alters the predicted play. It is easy to credibly reveal the ordinal preferences. Players cannot credibly reveal the intensity of preferences. Implications to anti-trust policy: Successful collusion often depend on the firms’ private preferences (e.g., market sharing agreement prefer to serve). Our findings strengthens the importance of not allowing even a brief form of explicit communication between oligopolistic competitors. E.g., 1997 series of FCC ascending auctions (Cramton & Schwartz, 2000). Heller & Kuzmics Renegotiation & Coordination March 2020 30 / 33
Related Literature (1) Other applications of renegotiation-proofness Repeated games (complete information), contracts with moral hazard. Hart & Tirole (1988); van Damme (1989); Bernheim & Ray (1989); Evans & Maskin (1989); Forges (1994); Wen (1996), Maestri (2017); Strulovici (2017). Private values in coordination games without communication Stylized result: Inefficient interior equilibria are stable if the type’s density is U-shaped. Details Kreps & Fudenberg (1993); Ellison & Fudenberg (2000); Sandholm (2007); Jelnov, Tauman & Zhao (2018) . Heller & Kuzmics Renegotiation & Coordination March 2020 31 / 33
Related Literature (1) Other applications of renegotiation-proofness Repeated games (complete information), contracts with moral hazard. Hart & Tirole (1988); van Damme (1989); Bernheim & Ray (1989); Evans & Maskin (1989); Forges (1994); Wen (1996), Maestri (2017); Strulovici (2017). Private values in coordination games without communication Stylized result: Inefficient interior equilibria are stable if the type’s density is U-shaped. Details Kreps & Fudenberg (1993); Ellison & Fudenberg (2000); Sandholm (2007); Jelnov, Tauman & Zhao (2018) . Heller & Kuzmics Renegotiation & Coordination March 2020 31 / 33
Related Literature (2) Communication in coordination games with public values Stylised result: Pareto-dominant outcome is selected if there are unused messages (secret handshake argument). Wärneryd, (1992); Schlag (1993); Sobel (1993); Kim & Sobel (1995 ); Bhaskar (1998); Banerjee & Weibull (2000); Hurkens & Schlag (2003) . Communication in stag-hunt games with private values Cheap-talk allows the Pareto-dominant outcome to be played with high probability (Baliga & Sjostrom, 2004). Heller & Kuzmics Renegotiation & Coordination March 2020 32 / 33
Related Literature (2) Communication in coordination games with public values Stylised result: Pareto-dominant outcome is selected if there are unused messages (secret handshake argument). Wärneryd, (1992); Schlag (1993); Sobel (1993); Kim & Sobel (1995 ); Bhaskar (1998); Banerjee & Weibull (2000); Hurkens & Schlag (2003) . Communication in stag-hunt games with private values Cheap-talk allows the Pareto-dominant outcome to be played with high probability (Baliga & Sjostrom, 2004). Heller & Kuzmics Renegotiation & Coordination March 2020 32 / 33
Conclusion Novel simple family of equilibria in coordination games with private values & cheap-talk: (1) agents always coordinate, (2) each agent states his preferred outcome (& nothing else), (3) agents coordinate on a mutually-preferred outcome (if exists). We show that it satisfies various appealing properties: Main result: strategy is in the family iff it is renegotiation-proof equilibrium. 1 Behavior is independent of the distribution of types. 2 Interim Pareto-efficiency. 3 Ex-ante payoff: best among coordinated eq., improves babbling equilibria. 4 Stable WRT various perturbations. 5 Heller & Kuzmics Renegotiation & Coordination March 2020 33 / 33
Backup Slides Heller & Kuzmics Renegotiation & Coordination March 2020 34 / 33
Pedestrian Traffic (Erving Goffman, 1971, Relations in Public, Ch. 1, P. 6) Back ”Take, for example, techniques that pedestrians employ in order to avoid bumping into one another. These seem of little significance. However, there are an appreciable number of such devices; they are constantly in use and they cast a pattern on street behavior. Street traffic would be a shambles without them.” Heller & Kuzmics Renegotiation & Coordination March 2020 35 / 33
Formal Definition of σ L = ( µ ∗ , ξ L ) and σ R = ( µ ∗ , ξ R ) Fix m L , m R ∈ M . Stage 1: Each agent states his preferred outcome: u ≤ 0 . 5 m L µ ∗ ( u ) = u > 0 . 5 m R Stage 2: ξ L : Play L iff at least one player prefers L. ξ R : Play R iff at least one player prefers R. m = m ′ = m R m = m ′ = m L R L ξ L ( m , m ′ ) = ξ R ( m , m ′ ) = L otherwise , R otherwise . Heller & Kuzmics Renegotiation & Coordination March 2020 36 / 33
Formal Definition of σ L = ( µ ∗ , ξ L ) and σ R = ( µ ∗ , ξ R ) Fix m L , m R ∈ M . Stage 1: Each agent states his preferred outcome: u ≤ 0 . 5 m L µ ∗ ( u ) = u > 0 . 5 m R Stage 2: ξ L : Play L iff at least one player prefers L. ξ R : Play R iff at least one player prefers R. m = m ′ = m R m = m ′ = m L R L ξ L ( m , m ′ ) = ξ R ( m , m ′ ) = L otherwise , R otherwise . Heller & Kuzmics Renegotiation & Coordination March 2020 36 / 33
Binary Communication – Formal Definition: Back Definition Let β σ ( m ) the expected probability that a player (who follows strategy σ ) plays L conditional on the opponent sending message m . Definition (Binary communication) � � β σ ( m ) ∈ β , β for any message m . β σ ( m ) = β if any type u < 0 . 5 sends message m . β σ ( m ) = β if any type u > 0 . 5 sends message m . Heller & Kuzmics Renegotiation & Coordination March 2020 37 / 33
Binary Communication – Formal Definition: Back Definition Let β σ ( m ) the expected probability that a player (who follows strategy σ ) plays L conditional on the opponent sending message m . Definition (Binary communication) � � β σ ( m ) ∈ β , β for any message m . β σ ( m ) = β if any type u < 0 . 5 sends message m . β σ ( m ) = β if any type u > 0 . 5 sends message m . Heller & Kuzmics Renegotiation & Coordination March 2020 37 / 33
Neutral Stability Back Definition (Maynard Smith & Price, 1973) Equilibrium strategy σ is neutrally stable iff it achieves a higher payoff against any best-reply strategy σ ′ , i.e., π ( σ ′ , σ ) = π ( σ , σ ) ⇒ π ( σ , σ ′ ) ≥ π ( σ ′ , σ ′ ) . Proposition Any renegotiation-proof Equilibrium Strategy σ = ( µ , ξ ) is neutrally stable. Sketch of Proof. σ ′ = ( µ ′ , ξ ′ ) is a best-reply against σ ⇒ ξ ′ = ξ , and µ ′ ≈ µ : µ and µ ′ may differ only WRT equivalent messages or the behavior of u = 0 . 5, i.e., µ u ( M L ) = µ ′ u ( M L ) and µ u ( M R ) = µ ′ u ( M R ) for any u � = 0 . 5. Heller & Kuzmics Renegotiation & Coordination March 2020 38 / 33
Neutral Stability Back Definition (Maynard Smith & Price, 1973) Equilibrium strategy σ is neutrally stable iff it achieves a higher payoff against any best-reply strategy σ ′ , i.e., π ( σ ′ , σ ) = π ( σ , σ ) ⇒ π ( σ , σ ′ ) ≥ π ( σ ′ , σ ′ ) . Proposition Any renegotiation-proof Equilibrium Strategy σ = ( µ , ξ ) is neutrally stable. Sketch of Proof. σ ′ = ( µ ′ , ξ ′ ) is a best-reply against σ ⇒ ξ ′ = ξ , and µ ′ ≈ µ : µ and µ ′ may differ only WRT equivalent messages or the behavior of u = 0 . 5, i.e., µ u ( M L ) = µ ′ u ( M L ) and µ u ( M R ) = µ ′ u ( M R ) for any u � = 0 . 5. Heller & Kuzmics Renegotiation & Coordination March 2020 38 / 33
Neutral Stability Back Definition (Maynard Smith & Price, 1973) Equilibrium strategy σ is neutrally stable iff it achieves a higher payoff against any best-reply strategy σ ′ , i.e., π ( σ ′ , σ ) = π ( σ , σ ) ⇒ π ( σ , σ ′ ) ≥ π ( σ ′ , σ ′ ) . Proposition Any renegotiation-proof Equilibrium Strategy σ = ( µ , ξ ) is neutrally stable. Sketch of Proof. σ ′ = ( µ ′ , ξ ′ ) is a best-reply against σ ⇒ ξ ′ = ξ , and µ ′ ≈ µ : µ and µ ′ may differ only WRT equivalent messages or the behavior of u = 0 . 5, i.e., µ u ( M L ) = µ ′ u ( M L ) and µ u ( M R ) = µ ′ u ( M R ) for any u � = 0 . 5. Heller & Kuzmics Renegotiation & Coordination March 2020 38 / 33
Message Function µ is Dominant (Given ξ ) Back Proposition Let ( µ , ξ ) be an RP equilibrium strategy. Then µ is a weakly dominant action, given σ , i.e., : π (( µ , σ ) , (( µ ′ , σ ))) ≥ π ((˜ µ , σ ) , (( µ ′ , σ ))) for any µ ′ , ˜ µ ∈ ∆( M ) . Sketch of proof. The impact of an agent’s action is its effect on the probability that the opponent plays L . µ u maximizes this probability for any u < 0 . 5, and minimizes this probability for any u > 0 . 5 ⇒ µ is a weakly dominant action function. Heller & Kuzmics Renegotiation & Coordination March 2020 39 / 33
Message Function µ is Dominant (Given ξ ) Back Proposition Let ( µ , ξ ) be an RP equilibrium strategy. Then µ is a weakly dominant action, given σ , i.e., : π (( µ , σ ) , (( µ ′ , σ ))) ≥ π ((˜ µ , σ ) , (( µ ′ , σ ))) for any µ ′ , ˜ µ ∈ ∆( M ) . Sketch of proof. The impact of an agent’s action is its effect on the probability that the opponent plays L . µ u maximizes this probability for any u < 0 . 5, and minimizes this probability for any u > 0 . 5 ⇒ µ is a weakly dominant action function. Heller & Kuzmics Renegotiation & Coordination March 2020 39 / 33
ξ is a Neighborhood Invader Strategy (Given µ ) Back Definition (Cressman, 2010) Strict equilibrium ( x 1 , x 2 ) of an asymmetric game is neighborhood invader strategy iff there is ε > 0, such that for each x ′ 1 � = x 1 and x ′ 2 � = x 2 , then either π 1 ( x 1 x ′ 2 ) > π 1 ( x ′ 1 x ′ 2 ) or π 2 ( x ′ 1 x 2 ) > π 2 ( x ′ 1 x ′ 2 ) . Cressman (2010) adapts to asymmetric games Apaloo’s (1997) notion of NIS (which refines the notion of CSS, Eshel & Motro, 1981). Proposition Let ( µ , ξ ) be an RP eq. strategy, and let m , m ′ ∈ supp ( µ ) . Then ( ξ ( m , m ′ ) , ξ ( m ′ , m )) is a neighborhood invader strict equilibrium in Γ( F m , F m ′ ) . Heller & Kuzmics Renegotiation & Coordination March 2020 40 / 33
ξ is a Neighborhood Invader Strategy (Given µ ) Back Definition (Cressman, 2010) Strict equilibrium ( x 1 , x 2 ) of an asymmetric game is neighborhood invader strategy iff there is ε > 0, such that for each x ′ 1 � = x 1 and x ′ 2 � = x 2 , then either π 1 ( x 1 x ′ 2 ) > π 1 ( x ′ 1 x ′ 2 ) or π 2 ( x ′ 1 x 2 ) > π 2 ( x ′ 1 x ′ 2 ) . Cressman (2010) adapts to asymmetric games Apaloo’s (1997) notion of NIS (which refines the notion of CSS, Eshel & Motro, 1981). Proposition Let ( µ , ξ ) be an RP eq. strategy, and let m , m ′ ∈ supp ( µ ) . Then ( ξ ( m , m ′ ) , ξ ( m ′ , m )) is a neighborhood invader strict equilibrium in Γ( F m , F m ′ ) . Heller & Kuzmics Renegotiation & Coordination March 2020 40 / 33
Coordination Games with no Communication Literature Analysis of the no-communication case is a special case of Sandholm (2007). All equilibria are based on “fixed-point” cut-offs: Each value x ∗ = F ( x ∗ ) induces cut-off equilibrium: play L iff x ≤ x ∗ . always -L ( x ∗ = 1), always -R ( x ∗ = 0), possibly interior cut-offs x ∗ ∈ ( 0 , 1 ) . An equilibrium is dynamically stable iff f ( x ∗ ) < 1. If f ( 0 ) , f ( 1 ) > 1, then only inefficient interior cut-off equilibria are dynamically stable. Illustration Heller & Kuzmics Renegotiation & Coordination March 2020 41 / 33
Coordination Games with no Communication Literature Analysis of the no-communication case is a special case of Sandholm (2007). All equilibria are based on “fixed-point” cut-offs: Each value x ∗ = F ( x ∗ ) induces cut-off equilibrium: play L iff x ≤ x ∗ . always -L ( x ∗ = 1), always -R ( x ∗ = 0), possibly interior cut-offs x ∗ ∈ ( 0 , 1 ) . An equilibrium is dynamically stable iff f ( x ∗ ) < 1. If f ( 0 ) , f ( 1 ) > 1, then only inefficient interior cut-off equilibria are dynamically stable. Illustration Heller & Kuzmics Renegotiation & Coordination March 2020 41 / 33
Example Literature Heller & Kuzmics Renegotiation & Coordination March 2020 42 / 33
Example Literature With communication: π ( σ R , σ R ) = π ( σ L , σ L ) = 1 . 27. Heller & Kuzmics Renegotiation & Coordination March 2020 43 / 33
Any equilibrium ( x 1 , x 2 ) of Γ( F m 1 , F m 2 ) can be Pareto-improved Back x 1 , x 2 ≤ 0 . 5 ⇒ σ R Pareto dominates ( x 1 , x 2 ) : u i > 0 . 5 gains because he gets his maximal feasible payoff in σ R . u i < 0 . 5 gains because there’s a higher probability of the partner playing L & a higher probability of coordination. x 1 , x 2 ≥ 0 . 5 ⇒ σ L Pareto dominates ( x 1 , x 2 ) (analogous argument). x 1 < 0 . 5 < x 2 : x 1 < 0 . 5 is indifferent between the 2 actions ⇒ Agent 2 usually plays R . 0 . 5 < x 2 is indifferent between the 2 actions ⇒ Agent 1 usually plays L . Coordination probability < 0.5 ⇒ payoff of ( x 1 , x 2 ) < 0 . 5 ⇒ σ C Pareto dominates ( x 1 , x 2 ) .
Any equilibrium ( x 1 , x 2 ) of Γ( F m 1 , F m 2 ) can be Pareto-improved Back x 1 , x 2 ≤ 0 . 5 ⇒ σ R Pareto dominates ( x 1 , x 2 ) : u i > 0 . 5 gains because he gets his maximal feasible payoff in σ R . u i < 0 . 5 gains because there’s a higher probability of the partner playing L & a higher probability of coordination. x 1 , x 2 ≥ 0 . 5 ⇒ σ L Pareto dominates ( x 1 , x 2 ) (analogous argument). x 1 < 0 . 5 < x 2 : x 1 < 0 . 5 is indifferent between the 2 actions ⇒ Agent 2 usually plays R . 0 . 5 < x 2 is indifferent between the 2 actions ⇒ Agent 1 usually plays L . Coordination probability < 0.5 ⇒ payoff of ( x 1 , x 2 ) < 0 . 5 ⇒ σ C Pareto dominates ( x 1 , x 2 ) .
Any equilibrium ( x 1 , x 2 ) of Γ( F m 1 , F m 2 ) can be Pareto-improved Back x 1 , x 2 ≤ 0 . 5 ⇒ σ R Pareto dominates ( x 1 , x 2 ) : u i > 0 . 5 gains because he gets his maximal feasible payoff in σ R . u i < 0 . 5 gains because there’s a higher probability of the partner playing L & a higher probability of coordination. x 1 , x 2 ≥ 0 . 5 ⇒ σ L Pareto dominates ( x 1 , x 2 ) (analogous argument). x 1 < 0 . 5 < x 2 : x 1 < 0 . 5 is indifferent between the 2 actions ⇒ Agent 2 usually plays R . 0 . 5 < x 2 is indifferent between the 2 actions ⇒ Agent 1 usually plays L . Coordination probability < 0.5 ⇒ payoff of ( x 1 , x 2 ) < 0 . 5 ⇒ σ C Pareto dominates ( x 1 , x 2 ) .
Any equilibrium ( x 1 , x 2 ) of Γ( F m 1 , F m 2 ) can be Pareto-improved Back x 1 , x 2 ≤ 0 . 5 ⇒ σ R Pareto dominates ( x 1 , x 2 ) : u i > 0 . 5 gains because he gets his maximal feasible payoff in σ R . u i < 0 . 5 gains because there’s a higher probability of the partner playing L & a higher probability of coordination. x 1 , x 2 ≥ 0 . 5 ⇒ σ L Pareto dominates ( x 1 , x 2 ) (analogous argument). x 1 < 0 . 5 < x 2 : x 1 < 0 . 5 is indifferent between the 2 actions ⇒ Agent 2 usually plays R . 0 . 5 < x 2 is indifferent between the 2 actions ⇒ Agent 1 usually plays L . Coordination probability < 0.5 ⇒ payoff of ( x 1 , x 2 ) < 0 . 5 ⇒ σ C Pareto dominates ( x 1 , x 2 ) .
Sketch of Proof of Main Result (2) Back If m , m ′ ∈ M L agents coordinate on L ; If m , m ′ ∈ M R agents coordinate on R . Otherwise, players renegotiate to the preferred outcome. Agents always coordinate after observing m ∈ M L and m ′ ∈ M R . Payoff of interior cutoff equilibrium ( x ∗ , y ∗ ) when one players prefer L and the opponent prefers R is at most 0.5. Agents renegotiate to coordinate on each outcome with probability 0.5. Agents with u ≤ 0 . 5 are indifferent between m , m ′ ∈ M L ⇒ the average probability to coordinate on L is the same after sending m and m ′ .
Sketch of Proof of Main Result (2) Back If m , m ′ ∈ M L agents coordinate on L ; If m , m ′ ∈ M R agents coordinate on R . Otherwise, players renegotiate to the preferred outcome. Agents always coordinate after observing m ∈ M L and m ′ ∈ M R . Payoff of interior cutoff equilibrium ( x ∗ , y ∗ ) when one players prefer L and the opponent prefers R is at most 0.5. Agents renegotiate to coordinate on each outcome with probability 0.5. Agents with u ≤ 0 . 5 are indifferent between m , m ′ ∈ M L ⇒ the average probability to coordinate on L is the same after sending m and m ′ .
Sketch of Proof of Main Result (2) Back If m , m ′ ∈ M L agents coordinate on L ; If m , m ′ ∈ M R agents coordinate on R . Otherwise, players renegotiate to the preferred outcome. Agents always coordinate after observing m ∈ M L and m ′ ∈ M R . Payoff of interior cutoff equilibrium ( x ∗ , y ∗ ) when one players prefer L and the opponent prefers R is at most 0.5. Agents renegotiate to coordinate on each outcome with probability 0.5. Agents with u ≤ 0 . 5 are indifferent between m , m ′ ∈ M L ⇒ the average probability to coordinate on L is the same after sending m and m ′ .
Illustration: Stable Pair of Thresholds Robustness No Communication If ξ ( m 1 , m 2 ) , ξ ( m 2 , m 1 ) are slightly perturbed, then best-reply dynamics induce agents to converge back to the eq. thresholds. (picture adapted from Sandholm, 2010)
General non Stag Hunt Coordination Games Back Type is a tuple ( u LL , u LR , u RL , u RR ) describing the payoff matrix. Feasible types: min( u LL , u RR ) > max( u RL , u LR ) (any non stag hunt coordination game). Asymmetric games are allowed: Different distributions of types F 1 and F 2 . Adaptation of RP strategies: each agent reports if u LL > u RR or u LL < u RR . For asymmetric games, the set of RP eq. strategies is 2-dimensional: σ α 1 , α 2 , where α i ∈ [ 0 , 1 ] denotes the probability of coordinating on L when the agent of population i prefers L , and the other agent prefers R . Heller & Kuzmics Renegotiation & Coordination March 2020 47 / 33
General non Stag Hunt Coordination Games Back Type is a tuple ( u LL , u LR , u RL , u RR ) describing the payoff matrix. Feasible types: min( u LL , u RR ) > max( u RL , u LR ) (any non stag hunt coordination game). Asymmetric games are allowed: Different distributions of types F 1 and F 2 . Adaptation of RP strategies: each agent reports if u LL > u RR or u LL < u RR . For asymmetric games, the set of RP eq. strategies is 2-dimensional: σ α 1 , α 2 , where α i ∈ [ 0 , 1 ] denotes the probability of coordinating on L when the agent of population i prefers L , and the other agent prefers R . Heller & Kuzmics Renegotiation & Coordination March 2020 47 / 33
Extreme Types with Dominant Actions Back The set of feasible types is [ a , b ] , where a < 0 and b > 1. Extreme types: u < 0 ( L is dominant) or u > 1 ( R is dominant). Assumption: extreme types are minority: F ( 0 ) < 0 . 5 · F ( 0 . 5 ) , 1 − F ( 1 ) < 0 . 5 · ( 1 − F ( 0 . 5 )) . and F ( 0 ) Essentially Unique RP equilibrium strategy σ α , where α ≡ F ( 0 )+( 1 − F ( 1 )) . σ C is renegotiation-proof in the symmetric case ( F ( 0 ) = 1 − F ( 1 ) ). Heller & Kuzmics Renegotiation & Coordination March 2020 48 / 33
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