Viable Nash Equilibria: Formation and Defection Ehud Kalai See the - PowerPoint PPT Presentation

One World Mathematical Game Theory seminar May 19, 2020, 15:00 CEST at https://zoom.us/j/241150956?pwd=clc4L1I5M3BYTXEvQnErOVBtRFI0UT09 Viable Nash Equilibria: Formation and Defection Ehud Kalai See the complete paper at Research Gate: https://www.researchgate.net/publication/339486928_VIABLE_NASH_EQUILIBRIA_FORMATION_AND_DEFECTION_FEB_2020 1

Viable Nash Equilibria: Formation & Defection Ehud Kalai Motivation: To be credible, economic applications should use only Nash equilibria that are viable. Simple dual indices: formation index, 𝑮(𝜌) = the # of players that “can form 𝜌 ,” defection index, 𝑬(𝜌) = the # of defectors that “ 𝜌 can sustain.” Surprisingly, these simple indices: 1. predict the performance of Nash equilibria in social systems and lab experiments, i.e. assess viability , 2. they also uncover new basic properties of Nash equilibria that have eluded game theory refinements. 2

Viable Nash Equilibria: Formation & Defection Ehud Kalai Motivation: To be credible, economic analysis should restrict itself to the use of only those Nash equilibria The broader goal : Develop theoretical tools to that are viable. Simple minimal departure from Nash: answer behavioral questions. • Stay with anonymous ordinal defections. I study simple dual indices to assess equil viability: Similar to • Only replace Nash’s assumption that a formation index, 𝐺(𝜌), and a defection index, 𝐸(𝜌) . 𝑭(𝒀) & 𝑻𝑬(𝒀) that assess “no opponents defect” by viability of investments 𝒀 , 𝜺 = “ the # of potential defectors .” Surprisingly, these simple indices: 𝑮(𝝆) & 𝑬(𝝆) assess ∴ attribute all new observations to 𝜺 . viability of equilibria 𝝆 . 1. predict the performance of Nash equilibria in social systems and lab experiments, i.e. viability. Avoid game theory refinements for: • Duality: 𝑮 𝝆 + 𝑬 𝝆 = n broad applicability. • 2. They also the # of players. simple best-response computations. uncover new properties of Nash equilibria and stability issues that have eluded gt refinements. 3

Viable Nash Equilibria Ehud Kalai Related earlier work Motivation: Viable Nash equilibria provide good • In theory : 𝑬 𝜌 = the level of subgame perfection understanding of functioning social systems, whereas in play with revisions, as in Kalai & Neme (1992). un viable Nash equilibria are often contrived, and have • In applications : 𝑬 originated in distributive computing , the appearance of useless theory. adopted to implementation theory by Eliaz (2002), I study simple dual indices to assess equilibrium Abraham et al. (2006), and Gradwohl and Reingold (2014). viability: a formation index, F( π ), and a deterrence • 𝑮 is new. index, D( π ). • This paper presents theory and applications of Surprisingly, despite their simplicity these indices: more extensive properties of 𝑬 and the new index 𝑮. 1. identifies new properties of Nash equilibria and stability issues beyond game theory refinements and 2. provide insights for the viability of Nash equil in functioning social systems and lab experiments. 4

Viable Nash Equilibria Viable Nash Equilibria Viable Nash Equilibria The game theory story: Ehud Kalai Ehud Kalai Rational players play an equilibrium π , but Ehud Kalai concerned about defections by opponents. A solution of a game is viable if its play is credible, considering A solution of a game is viable if its play is credible, considering Motivation: Viable Nash equilibria provide good Examples: the broad context in which the game is played. the broad context in which the game is played. understanding of functioning social systems, whereas • Faulty opponents , irrational, unpredictable, see Eliaz (2002). un viable Nash equilibria are often contrived, and have Are Nash equilibria viable? My answer: some are, some not. Answer: some are, some not. • Coalitions of rational defectors. the appearance of useless theory. • Incomplete game specifications : e.g., threats, bribes, Will present: Two (dual) indices to assess viability of equilibrium π I study simple dual indices to assess equilibrium reputation for future play, misspecified payoffs,… 1 . Defection-deterrence, 𝑬(π) , 1 . Defection-deterrence, 𝑬(π) , viability: a formation index, F( π ), and a deterrence 𝑬 and 𝑮 : simple indices 𝑬 and 𝑮 : simple indices, the player-power needed to the player-power needed to index, D( π ). to explain social systems explain NE play in social undo π (focal sustainability). undo π ( ~ π sustainability as a A strategy is “highly viable,” if it is “dominant against many and lab experiments. Surprisingly, despite their simplicity these indices: systems and lab experiments. focal point until the play). defections.” Strong condition, yet observed in social 1. identifies new properties of Nash equilibria and The simplicity is a virtue. Their simplicity is the virtue. systems; more manageable than optimal Bayesian response. stability issues beyond game theory refinements and 2. provide insights for the viability of Nash equil in functioning social systems and lab experiments. 5 5 5

Lecture topics: Definitions and properties (1) Subjective viability assessments (1) Behavioral observations (1) Incomplete info game (1) Comparisons with standard GT (1) Rational coalitional defections (1/2) Forming/switching equilibrium (1/4) Implementations with faulty players (1/4) Viability in network games (1/4) Future research (1/2) Proposed experiment (1/4) 6

Definitions and properties 𝜌 - a fixed strategy profile of an 𝑜 -person strategic game Γ . Definition: D( 𝜌 ) ≡ the minimal number of defections from 𝜌 needed to construct a 𝜌′ to which 𝜌 is not a best response. D( 𝜌 ) measures the confidence in individual strategies: With any # of defections < D( 𝜌 ), everybody’s π i is optimal. resilience in Abraham et al (2006) Equivalent definition: D( 𝜌 ) ≡ the maximal 𝑒 s.t. “ 𝜌 deters 𝑒 potential defectors .” That is: in any subgame played by 𝑒 (potentially defecting) players, 𝐻 , 𝜌 is a dominant-strategy eqm. (Assuming that the remaining n-d players in 𝐻 𝑑 are 𝜌 loyalists). Beyond Nash’s deterrence of any one potential defector, 𝜌 strongly deters defection of any D( 𝜌 ) potential defectors. 7

Definitions and properties Next: A dual restatement, stated through the number of loyalist: (Dual) Definition of the formation index 𝐺 𝜌 ≡ The minimal 𝑚 s.t. for any group of 𝑚 -loyalists, 𝜌 𝑗 is a dominant strategy for any player 𝑗 outside the group. Any 𝐺(𝜌) loyalists strongly induce the play of 𝜌 on the rest. Duality 𝐸(𝜌) + 𝐺(𝜌) = 𝑜 , each useful in different applications. 8

Definitions and properties As in Eliaz (2002) : Tolerance of 𝐸 𝜌 − 1 faulty players: If 𝐸 𝜌 − 1 players are faulty (irrational and unpredictable), 𝜌 remains a Nash eqm of the non-faulty players. Thus, a new natural concept: Nash critical mass of π ≡ 𝑮 𝝆 + 1 . bounded “Small worlds” If a group 𝐻 has at least 𝑮 𝝆 + 1 players, then 𝜌 is a Nash eqm for 𝐻 no matter what the others play. Nash/dominance complementarity: For 𝑑 = 0,1 … , 𝑜 − 1 : 𝜌 is a Nash eqm for every group of 𝑑 + 1 players (regardless of the actions of the others) iff 𝜌 is a dominant-strategy eqm for any group of 𝑜 − 𝑑 players (when the others are 𝜌 loyalists). 9

Definitions and properties Nash equilibria are rungs on a ladder of deterrence/dominance 𝐸 𝜌 = 0,1, … , n partitions all stgy profiles 𝜌 of n player games 𝑬 𝝆 = n iff 𝝆 is a dominant strgy eqm. 𝜌 deters any # defectors 𝐸 𝜌 = n-1 iff 𝜌 is a dominant strgy eqm, given 1 loyalist. 𝐸 𝜌 = n-2 iff 𝜌 is a dominant strgy eqm, given 2 loyalists. 𝐸 𝜌 = 2 iff 𝜌 deters up to 2 defectors iff 𝜌 deters single player defections, not more. 𝐸 𝜌 = 1 𝑬 𝝆 = 0 iff 𝝆 is not a Nash equilibirum 10

Example: Asymmetric small game The Party Line game . 3 𝐸 emocrats and 5 𝑆 epublicans, each chooses 𝐹 or 𝐺 . Payoffs: # opposite-party players you mis match. 𝑬𝒋𝒘 isive eqm: 𝐸𝑓𝑛 s choose 𝐺 ; 𝑆𝑓𝑞 s choose 𝐹 . 𝐸 𝐸𝑗𝑤 = min 2,3 = 2 Why? 𝐺(𝐸𝑗𝑤) = 8 − 2 = 6 11

Lecture topics: Definitions and properties (1) Subjective viability assessments (1) Behavioral observations (1) Incomplete info game (1) Comparisons with standard GT (1) Rational coalitional defections (1/2) Forming/switching equilibrium (1/4) Implementations with faulty players (1/4) Viability in network games (1/4) Future research (1/2) Proposed experiment (1/4) 12

𝐸 assesses equil sustainability High sustainability: Language Matching 𝑜 = 200𝑁 players, choose a language. Payoff: # of opponents you match. Equil: all English, 𝒃𝑭 . 𝑬 𝒃𝑭 = 𝟐𝟏𝟏𝑵. Low sustainability: Random Language 𝑜 = 200𝑁 players, choose a language. Payoff: # of opponents you match. Equil : everybody flips a coin English/French 𝒇𝑫𝑭/𝑮 𝑬 𝒇𝑫𝑭/𝑮 = 𝟐. 13

Viability assessment Sustainability D ( π ) 200M all 100M English . . . . 2 1 0 14