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

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Viable Nash Equilibria: Formation and Defection

Ehud Kalai

One World Mathematical Game Theory seminar May 19, 2020, 15:00 CEST at

https://zoom.us/j/241150956?pwd=clc4L1I5M3BYTXEvQnErOVBtRFI0UT09 See the complete paper at Research Gate:

https://www.researchgate.net/publication/339486928_VIABLE_NASH_EQUILIBRIA_FORMATION_AND_DEFECTION_FEB_2020

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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.

Viable Nash Equilibria: Formation & Defection

Ehud Kalai

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Motivation: To be credible, economic analysis should restrict itself to the use of only those Nash equilibria that are viable.

Viable Nash Equilibria: Formation & Defection

Ehud Kalai

I study simple dual indices to assess equil viability: a formation index, 𝐺(𝜌), and a defection index, 𝐸(𝜌). Surprisingly, these simple indices:

• 1. predict the performance of Nash equilibria in

social systems and lab experiments, i.e. viability.

• 2. They also

uncover new properties of Nash equilibria and stability issues that have eluded gt refinements.

The broader goal: Develop theoretical tools to answer behavioral questions. Similar to 𝑭(𝒀) & 𝑻𝑬(𝒀) that assess viability of investments 𝒀, 𝑮(𝝆) & 𝑬(𝝆) assess viability of equilibria 𝝆. Simple minimal departure from Nash:

• Stay with anonymous ordinal defections.
• Only replace Nash’s assumption that

“no opponents defect” by 𝜺 = “the # of potential defectors.” ∴ attribute all new observations to 𝜺. Avoid game theory refinements for:

• broad applicability.
• simple best-response computations.

Duality: 𝑮 𝝆 + 𝑬 𝝆 = n the # of players.

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Motivation: Viable Nash equilibria provide good understanding of functioning social systems, whereas unviable Nash equilibria are often contrived, and have the appearance of useless theory.

Viable Nash Equilibria

Ehud Kalai

I study simple dual indices to assess equilibrium viability: a formation index, F(π), and a deterrence index, D(π). Surprisingly, despite their simplicity these indices:

• 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. Related earlier work

• In theory: 𝑬 𝜌 = the level of subgame perfection

in play with revisions, as in Kalai & Neme (1992).

• In applications: 𝑬 originated in distributive computing,

adopted to implementation theory by Eliaz (2002), Abraham et al. (2006), and Gradwohl and Reingold (2014).

• 𝑮 is new.
• This paper presents theory and applications of

more extensive properties of 𝑬 and the new index 𝑮.

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Motivation: Viable Nash equilibria provide good understanding of functioning social systems, whereas unviable Nash equilibria are often contrived, and have the appearance of useless theory.

Viable Nash Equilibria

Ehud Kalai

I study simple dual indices to assess equilibrium viability: a formation index, F(π), and a deterrence index, D(π). Surprisingly, despite their simplicity these indices:

• 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.

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A solution of a game is viable if its play is credible, considering the broad context in which the game is played.

• 1. Defection-deterrence, 𝑬(π),

the player-power needed to undo π (focal sustainability).

Viable Nash Equilibria Ehud Kalai

Are Nash equilibria viable? My answer: some are, some not. Will present: Two (dual) indices to assess viability of equilibrium π 𝑬 and 𝑮: simple indices, explain NE play in social systems and lab experiments. Their simplicity is the virtue.

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A solution of a game is viable if its play is credible, considering the broad context in which the game is played.

• 1. Defection-deterrence, 𝑬(π),

the player-power needed to undo π (~ π sustainability as a focal point until the play).

Viable Nash Equilibria Ehud Kalai

Answer: some are, some not. 𝑬 and 𝑮: simple indices to explain social systems and lab experiments. The simplicity is a virtue.

The game theory story:

Rational players play an equilibrium π, but concerned about defections by opponents. A strategy is “highly viable,” if it is “dominant against many defections.” Strong condition, yet observed in social systems; more manageable than optimal Bayesian response.

• Faulty opponents, irrational, unpredictable, see Eliaz (2002).

Examples:

• Coalitions of rational defectors.
• Incomplete game specifications: e.g., threats, bribes,

reputation for future play, misspecified payoffs,…

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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)

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𝜌 - a fixed strategy profile of an 𝑜-person strategic game Γ. D(𝜌)≡ the minimal number of defections from 𝜌 needed to construct a 𝜌′ to which 𝜌 is not a best response. 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 𝐻𝑑 are 𝜌 loyalists).

Equivalent definition: D(𝜌) measures the confidence in individual strategies: With any # of defections < D(𝜌), everybody’s πi is optimal. Beyond Nash’s deterrence of any one potential defector, 𝜌 strongly deters defection of any D(𝜌) potential defectors.

Definitions and properties

resilience in Abraham et al (2006)

Definition:

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(Dual) Definition of the formation index 𝐺 𝜌 ≡ The minimal 𝑚 s.t. for any group of 𝑚-loyalists, 𝜌𝑗 is a dominant strategy for any player 𝑗 outside the group. Any 𝐺(𝜌) loyalists strongly induce the play of 𝜌 on the rest.

Definitions and properties

Next: A dual restatement, stated through the number of loyalist: 𝐸(𝜌) + 𝐺(𝜌) = 𝑜, each useful in different applications. Duality

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If 𝐸 𝜌 − 1 players are faulty (irrational and unpredictable), 𝜌 remains a Nash eqm of the non-faulty players.

Definitions and properties

Tolerance of 𝐸 𝜌 − 1 faulty players: Nash critical mass of π ≡ 𝑮 𝝆 + 1. Thus, a new natural concept:

As in Eliaz (2002): 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. 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). Nash/dominance complementarity:

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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

iff 𝜌 deters single player defections, not more. iff 𝜌 deters up to 2 defectors 𝜌 deters any # defectors 𝑬 𝝆 = n iff 𝝆 is a dominant strgy eqm. 𝑬 𝝆 = 0 iff 𝝆 is not a Nash equilibirum 𝐸 𝜌 = 1 𝐸 𝜌 = 2 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.

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The Party Line game. 3 𝐸emocrats and 5 𝑆epublicans, each chooses 𝐹 or 𝐺. Payoffs: # opposite-party players you mismatch. 𝑬𝒋𝒘isive eqm: 𝐸𝑓𝑛s choose 𝐺; 𝑆𝑓𝑞s choose 𝐹. 𝐸 𝐸𝑗𝑤 = min 2,3 = 2 𝐺(𝐸𝑗𝑤) = 8 − 2 = 6

Example: Asymmetric small game

Why?

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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)

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𝑜 = 200𝑁 players, choose a language. Payoff: # of opponents you match. Equil: all English, 𝒃𝑭.

𝐸 assesses equil sustainability

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

Viability assessment

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Sustainability D(π)

all English

. . . . 2 1 200M 100M

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𝑜 = 200𝑁, subscribe or not, subscription costs $9.99 Easy formation: New Communication Network Equil: all subscribe, 𝒃𝑻𝒗𝒄. 𝑮(𝒃𝑻𝒗𝒄) = 𝟐𝟏. 𝐸 𝒃𝑻𝒗𝒄 = 200𝑁 − 10.

sustaining 𝐺 assesses equil formation

𝑜 = 200𝑁, choose a language. Payoff: # of opponents you match. 𝑬 𝒃𝑭 = 100𝑁. Equil: all choose English, 𝑏𝐹. Difficult formation: Language Matching all choose French, 𝒃𝑮. 𝐸(𝑏𝐺) Non-sub’er = 0 Sub’er = # (other sub′ers) − 9.99. Payoffs = 𝑮(𝒃𝑮) = 𝟐𝟏𝟏𝑵

Viability assessment

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Formation difficulty F(π) Sustainability D(π)

Subjective

Viability Assessment

all English

. . . 2 1 1 2 . . . 200M 200M 100M 100M 10 200M-10

Viability assessment

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Formation difficulty F(π) Sustainability D(π)

Subjective

Viability Assessment

all French all English

. . . 2 1 1 2 . . . 200M 200M 100M 100M 10 200M-10

Viability assessment

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Formation difficulty F(π) Sustainability D(π)

Subjective

Viability Assessment

all French/English all English/French

. . . 2 1 1 2 . . .

High viability Low viability

200M 200M 100M 100M 10 200M-10

?

Viability assessment

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Formation difficulty F(π) Sustainability D(π)

Subjective

Viability Assessment

all French/English all English/French

. . . 2 1 1 2 . . .

High viability Low viability

200M 200M 100M 100M 10 200M-10

Viable if already exists

Viability assessment

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Formation difficulty F(π) Sustainability D(π)

Subjective

Viability Assessment

all French/English all English/French

. . . 2 1 1 2 . . .

High viability Low viability all English all French

200M 200M 100M 100M 10 200M-10

In US:

Viability assessment

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Formation difficulty F(π) Sustainability D(π)

Subjective

Viability Assessment

all French/English all English/French

. . . 2 1 1 2 . . .

High viability Low viability all French all English

200M 200M 100M 100M 10 200M-10

In France:

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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)

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𝑜 = 200𝑁 choose a language. Payoff: number of opponents you match. Def deterrence: 𝑬(𝒇𝑭) = 𝟐𝟏𝟏𝑵. Equil: all English, eE. recall Language-matching game Many similar social systems

• All English, all Spanish, all Mandarin… .
• All use dollars, all use euros, … .
• All use the metric system, all use U.S. measurements… .
• All use the Qwerty keyboard, all drive on the right, … .
• All subscribe to Facebook, to Twitter, to Zoom,…

Viability in functioning systems

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Nobody confess in the Confession game. All judges submit zero score in the Beauty-Contest game. All reporting to work in a simple production line. Mixed strategies equilibrium in coordination game. Centralized trade, Centralized communication…

See experimental results in Nagel (1995) and follow up papers. See O'Neill (1987) and follow up papers for experimental results. See Mishina (1992), "Toyota Motor Manufacturing, U.S.A., Inc.". Papers in finance and in network communication design. Non-disclosure agreements in bio and high tech companies.

Social avoidance of unviable equilibria

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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)

Why members of the same political party repeat the same talking points again and again?

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Example with Incomplete Information: Duplicating signals

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A signaling game S = {α, β}: possible states, pos actions, pos recommendations. Players: 1000 DMs; 3 Recommenders: 2 Honest and 1 Malicious. Equil 𝝆: RH = θ; 𝑆𝑁 = 𝜄𝑑; DMs: majority recom’n. 𝑬 𝝆 = 𝟐.

• Political parties duplicate recom’ders and talking points to

increase the viability of a good equilibrium.

• Losers vote to decrease the viability of a bad equilibrium.

Identical outcomes & Info structures at 𝝆, 𝝆’, 𝝆’’, but diff 𝐸s

• Recom’ders know true state, θ∊S; each rec’ends action Ri ∊S.
• DMs know the majority recom’n; each chooses action Ai ∊S.

Payoffs: DMs and Honest Recommenders = #(Ai = θ). Malicious Recommender = #(𝐵𝑗 ≠ 𝜄). If 𝑆𝑁

′ = 𝜄, then 𝑬 𝝆′ = 𝟑 (>1).

• If # of honest recom’ders 4, then 𝑬 𝝆′′ = 𝟑 (>1).
• Why does 𝑆𝑁 = 𝜄𝑑?

Playing for 𝐸 values, in addition to payoffs

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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)

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Comparisons with gt refinements in a confession game

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some confess→ each gets 10 yrs jail, but confessors only 3 yrs. 𝑜 = 36 crime participants, confess or not. nobody confesses → all go free. A Confession Game Equil: nobody confesses, 𝒐𝑫. 𝑬 𝒐𝑫 = 𝟐.

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Strong Nash. Proper, Perfect, Game theory laureates: nC is nC is coalition proof Despite passing the GT refinements, 𝑬 𝒐𝑫 = 𝟐 → 𝒐𝑫 is barely sustainable.

feasible payoffs

𝒗(𝒐𝑫) nC Pareto dominates all feasible payoffs

Game theory refinements vs the mafia

But mafias say nC is not viable.

Somebody will confess, if just for the fear that

• thers would.

Killing confessors should make nC a dominant strategy

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Comparison with stochastic stability in match the boss game.

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The boss 𝐶 and 𝑜 subordinates, each chooses a language. 𝐶’s payoff: 1 if he chooses 𝐹, 0 otherwise. Each subordinate’s payoff: 1 if she matches 𝐶, 0 otherwise. But still, 𝑓𝐹, is only minimally sustainable, 𝑬 𝒇𝑭 = 𝟐, Equil, all choose E, 𝒇𝑭, is stochastically stable a la Young (1993), KMR (1993), and other basin-of-attraction arguments. Why? The subordinates depend on the choice of

• ne player, B, who may

be unreliable.

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The boss 𝐶 and 𝑜 subordinates, each chooses a language. 𝐶’s payoff: 1 if he chooses 𝐹, 0 otherwise. Subordinate’s payoff: 1 if she matches 𝐶, 0 otherwise. But still, 𝑓𝐹, is only minimally sustainable, 𝑬 𝒇𝑭 = 𝟐, All choose E, 𝒇𝑭, is stochastically stable a la Young (1993), KMR (1993), and other basin-of-attraction arguments. Why? The subordinates depend on the choice of

• ne player, B, who may

be unreliable. The boss 𝐶 and 𝑜 subordinates, each chooses a language. 𝐶’s payoff: 1 if he chooses 𝐹, 0 otherwise. Subordinate’s payoff: 1 if she matches 𝐶, 0 otherwise. All choose E, 𝒇𝑭, is stochastically stable a la Young (1993), KMR (1993), and other basin-of-attraction arguments. The boss 𝐶 and 𝑜 subordinates, each chooses an alternative. 𝐶’s payoff: 1 if he chooses 𝐹, 0 otherwise. Each subordinate’s payoff: 1 if she matches 𝐶, 0 otherwise. All choose E, 𝑓𝐹, is stochastically stable a la Young (1993), KMR (1993), and other basin-of-attraction arguments. The boss 𝐶 and 𝑜 subordinates, each chooses an alternative. 𝐶’s payoff: 1 if he chooses 𝐹, 0 otherwise. Subordinate’s payoff: 1 if she matches 𝐶, 0 otherwise. All choose E, 𝑓𝐹, is stochastically stable a la Young (1993), KMR (1993), and other basin-of-attraction arguments. The boss 𝐶 and 𝑜 subordinates, each chooses an alternative. 𝐶’s payoff: 1 if he chooses 𝐹, 0 otherwise. Subordinate’s payoff: 1 if she matches 𝐶, 0 otherwise. All choose E, 𝑓𝐹, is stochastically stable a la Young (1993), KMR (1993), and other basin-of-attraction arguments. Improvement: have a committee of three bosses who prefer E, and n subordinates who wish to match most

• bosses. now:

𝑬 𝒇𝑭 = 𝟑 (> 𝟐). Evaluation by committees, not deans Replace dictators by politburos,…

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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)

Rideshare game

8 players, each can (1) take a taxi for $80, or (2) ride the bus. The Bus costs, $180, will be shared equally by the bus choosers 𝐸(𝑓𝑈) = 2 Equilibrium: Everybody takes a taxi, et.

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c = Cost of taxi 𝐸(𝑓𝑈) 8

180 22.5

7 1 Dominant How is the cost of the taxi related to D(e𝑈)? 𝐸 𝑓𝑈 = 0,1,…, 8, are all possible.

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# of bus riders x cost/rider $180/x cost of taxi $c Det Deterrence 𝐄(𝒇𝑼)) 180 < 𝑑 1 180 90 < 𝑑 ≤ 180 1 2 90 60 < 𝑑 ≤ 90 2 3 60 45 < 𝑑 ≤ 60 3 4 45 36 < 𝑑 ≤ 45 4 5 36 30 < 𝑑 ≤ 36 5 6 30 25.7 < 𝑑 ≤ 30 6 7 25.7 22.5 < 𝑑 ≤ 25.7 7 8 22.5 𝑑 ≤ 22.5 8

When taxi cost = $80, 𝑬 𝒇𝑼 = 2, why? A 2 player defection is a loss, cost go up from $80 to $90; but a 3 player defection may be a gain, cost go down from $80 to $60.

c = Cost of taxi 𝑬(𝒇𝑼)

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180 22.5

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As we vary the taxi cost, we obtain all values 𝐸(𝑓𝑈) = 0, … , 8.

These are rational incentives to defect, simple enough for bus riders, politicians,…

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# of bus riders x cost/rider $180/x cost of taxi $c Det Deterrence 𝐄(𝒇𝑼)) 180 < 𝑑 1 180 90 < 𝑑 ≤ 180 1 2 90 60 < 𝑑 ≤ 90 2 3 60 45 < 𝑑 ≤ 60 3 4 45 36 < 𝑑 ≤ 45 4 5 36 30 < 𝑑 ≤ 36 5 6 30 25.7 < 𝑑 ≤ 30 6 7 25.7 22.5 < 𝑑 ≤ 25.7 7 8 22.5 𝑑 ≤ 22.5 8

When taxi cost = $80, 𝑬 𝒇𝑼 = 2, why? A 2 player defection is a loss, cost go up from $80 to $90; but a 3 player defection may be a gain, cost go down from $80 to $60.

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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)

Likely switch to the bus

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Everybody bus, eB, has 𝑮 𝒇𝑪 = 𝟑. The Ridesharing game: 8 players, each can (1) take a taxi, for $80;

• r (2) ride the bus. The bus costs, $180, will be shared equally

among the bus choosers The small 𝑮 𝒇𝑪 is likely to cause a switch to the bus: EX.1 The bus company guarantees that the first 2 bus choosers will pay at most $75 each. Two will take it and make the bus the dominant strategy for the rest, all paying $22.5/rider. EX.2 Seeing the possible savings, 2 players initiate the switch to the bus on their own, without waiting for the bus company… .

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But you need to first convince 𝑮 𝒇𝑮 = 200M − 100M = 𝟐𝟏𝟏𝐍 players. Switching to all choosing French, 𝒇𝑮, in the language-choice game is unlikely. n=200M symmetric players choosing a language. 𝐸 𝑓𝑮 = 100M, highly sustainable Switching to “all choosing French” in the US is unlikely, it requires the commitment of 100M players.

Unlikely switch

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• In 1866, an act of the US Congress made it "lawful

throughout the United States of America to employ the weights and measures of the metric system...“

• In 1975 Congress enacted the Metric Conversion Act but

left the conversion voluntary. This attempt failed too, and the resistance to Metric continues. The high formation index requires a legal action to make people switch.

No Switch: A Sad Example

Attempts to switch the US out of its measurement system to metric keep failing.

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Recall: The Party Line game. 3 𝐸emocrats and 5 𝑆epublicans, each chooses 𝐹 or 𝐺. Payoffs: # opposite-party players you mismatch. 𝑬𝒋𝒘isive eqm: 𝐸𝑓𝑛s choose 𝐺; 𝑆𝑓𝑞s choose 𝐹. 𝐸 𝐸𝑗𝑤 = min 2,3 = 2 𝐺(𝐸𝑗𝑤) = 8 − 2 = 6

Example: Divisive equilibria

With larger parties, the divisive eqm becomes harder to form; but more sustainable, if formed. For example: If # Democrats = # Republicans = 1M, then 𝐸 𝐸𝑗𝑤 ≈ .5𝑁 and 𝐺 𝐸𝑗𝑤 ≈ 1.5𝑁

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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)

Equilibrium with “faulty players”

Abraham, I., D. Dolev, R. Gonen and J. Halpern, "Distributed Computing Meets Game Theory: Robust Mechanisms for Rational Secret Sharing and Multiparty Computation," the Proc. 25th ACM Symposium on Principles of Distributed Computing, 2006.

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Eliaz, K. "Fault Tolerant Implementation," REStud, 2002. Gradwohl and Reingold, ”Fault Tolerance in Large Games,” GEB, 2014.

Implementation with 𝑙 faulty players

Eliaz (2002): 𝑜-player implementation method, when 𝑙 unknown players may be faulty, i.e., irrational and unpredictable. Uses “𝑙-Fault Tolerant Nash Eq.”, 𝑙-FTNE: where the 𝑜 − 𝑙 rational players have the Nash incentives to play an equilibrium π, regardless of the strategies of the 𝑙 faulty unknown players. In our terminology, the # rational players ≥ Nash critical mass, or: π is a k-FTNE iff 𝑜 − 𝑙 ≥ Nash c.m. =𝐺 𝜌 + 1 = 𝑜 − 𝐸 𝜌 + 1

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DIFFERENCE Our potential defectors may be faulty as in Eliaz, but may also be rational. Allows new applications such as: bribing and/or threatening rational players, defections by rational coalitions, equilibrium formation/switching by rational players, ... ∴ 𝜌 facilitates Eliaz 𝑙-faulty-player implementation iff 𝑬(𝝆) > 𝒍.

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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)

Graph matching games

Equilibrium viability in social networks.

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Matching neighbors in a graph

Γ: a directed graph; V - set of 𝑜 vertices; 𝐹 - set of edges. The set of (out) neighbors of 𝑤, η 𝑤 ≡ 𝑥 ∈ 𝑊 𝑤, 𝑥 ∈ 𝐹 . The Γ matching game: V - the set of players; each choose a language. Payoff: the number of your neighbors you match. Equilibrium: everybody choosing 𝐹, 𝑓𝐹. THEOREM 𝑬 𝒇𝑭 : If ෠ 𝑊 ≠ ∅, then 𝑬(𝒇𝑭) = 𝐧𝐣𝐨

𝒘∈෡ 𝑾 |𝜽 𝒘 | 𝟑

+ 𝟐. = the strict majority of neighbors of the most vulnerable player, i.e., one with a minimal number of neighbors. Let ෠ 𝑊 = v η 𝑤 ≠ ∅}, the connected vertices. If ෠ 𝑊 = ∅, 𝑬(𝒇𝑭) = 𝑜.

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Matching neighbors in a graph

Γ: a directed graph; V - set of 𝑜 vertices; 𝐹 - set of edges. The set of (out) neighbors of 𝑤, η 𝑤 ≡ 𝑥 ∈ 𝑊 𝑤, 𝑥 ∈ 𝐹 . The Γ matching game: V - the set of players; each choose a language. Payoff: the number of your neighbors you match. Equilibrium: everybody choosing 𝐹, 𝑓𝐹. THEOREM 𝑬𝑬 𝒇𝑭 : If ෠ 𝑊 ≠ ∅, then 𝑬(𝒇𝑭) = 𝐧𝐣𝐨

𝒘∈෡ 𝑾 |𝜽 𝒘 | 𝟑

+ 𝟐. = the strict majority of neighbors of the most vulnerable player, i.e., one with a minimal number of neighbors. Let ෠ 𝑊 = v η 𝑤 ≠ ∅}, the connected vertices. If ෠ 𝑊 = ∅, 𝑬(𝒇𝑭) = 𝑜.

Matching neighbors in a graph

A currency match game with 𝑜 traders: An equilibrium, everybody chooses 𝐸, 𝑓𝐸. In a complete graph 𝑬 𝒇𝑬 ≈ 𝒐/𝟑 Trading through middleman: 𝑬 𝒇𝑬 = 𝟐. Decentralized equilibria are more sustainable Similarly for political, communications, supply chains, …

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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)

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Summary and Future research

Some open problems

• Axiomatize 𝐸, 𝐺, or other indices.
• Expand Nash existence theorem.
• Advance more refined indices.
• Non anonymous indices and equil formation.
• Sequential equilibrium formation.

That two simple theoretic indices, 𝑬 and 𝑮 explain and predict observed strategic behavior, suggests an important optimistic research agenda: Develop game theory tools to predict/explain actual strategic behavior.

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Nash’s Theorem restated: In any finite 𝑜 person game there exists a profile 𝜌 with 𝐸(𝜌)≥ 1. Important open problem: Since 𝐸(𝜌)=1 is often not sufficiently reliable: Find sufficient conditions for a game to have equilibrium 𝜌 with 𝐸(𝜌) ≥ 𝑙, for 𝑙 = 2 or more.

Extending Nash’s existence theorem

Northwestern University working paper by Deepanshu Vasal and Randall Berry, may be of interest.

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Limitations of 𝐸 (all shared with Nash eq)

Non anonymous indices: consider the identity of defectors and their position in the game. Needed Discontinuity: Unlike the lhs game, a single defection in the rhs game totally destroys the equilibrium. Needed Continuity: for a 𝐸 𝑓𝐹 that communication viability in the two populations.

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Equil formation & non-anonymous indices

A coalition G is a generator of π, if the play of π by G makes π a dominant strategy for the rest of the players, i.e., G forms 𝜌. Notice

• 1. Monotonicity: If G is a π -generator then so is G’, for G’⊇ G.

N is the unique maximal π - generator under containment.

• 2. Simple structure: There is a finite number of minimal (by

containment) generators, called roots of π. A coalition is a generator iff it contains a root.

• 3. Non-anonymous formation, (where you convince a root to

play π) may be more efficient than the anonymous formation (where you convince 𝐺 𝜌 players to play 𝜌).

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Illustration: the divisive equilibrium.

Recall: The divisive equilibrium in the Party Line game, 𝐸𝑗𝑤, in which 3 Ds choose F and 5 Rs choose E. Roots: The 3 Ds, The 5 Rs any group of 2Ds and 3Rs, {𝟑𝑬𝒕&𝟒𝑺𝒕}. Alternative ways to form 𝐸𝑗𝑤

• 1. Convince the 3 Ds to choose F.
• 2. Convince the 5 Rs to choose E.
• 3. Convince any 2 Ds and any 3Rs to play 𝐸𝑗𝑤.
• 4. Convince any 6 players to play 𝐸𝑗𝑤.

(1, 2, and 3) target the roots. (4) relies on the anonymous index 𝐺(𝐸𝑗𝑤) = 6. Non-anonymous methods require more computations.

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Illustration: the divisive equilibrium.

Recall: T divisive equilibrium in the Party Line game, 𝐸𝑗𝑤, in which the 3 Ds choose F and the 5 Rs choose E. Sequential equilibrium formation: for example, convince two Ds, this will convince all the Rs, which in turn will convince the third D. There are multiple paths for the order and choices of coalitions to convince, harder to compute … For example, the “genius of a dictator” is the skill to navigate a sequential formation process that lead to an equilibrium in which all the players obey her wishes.

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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)

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Proposed Experiment: 20-Player Participation Game with threshold 𝑢

𝑢 - a non-negative integer, 𝑢 ≥ 1. Simultaneously, each player chooses to participate or not. Non-participant’s payoff = $10. = ቊ$20, 0, Each participant’s payoff if the # of participants > 𝑢,

• therwise.

Everybody participates, eP, is a Nash eq for 𝑢 = 1, … , 19, with formation difficulty index 𝑮 𝒇𝑸 = 𝑢. Experiment question: would the number of participants decrease as the threshold 𝑢 is increased? Conjecture: Participation rates should decrease as t increases. Question: When do they ≈ 0?

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𝝆

𝑻𝟐 𝑻𝟑 𝑻𝟒

Basin of attraction of an equilibrium π ,

with 𝐸 𝜌 = 𝟐, in a 3 player game.

A 1 dimensional cross in a 3 dimension strategy space. domination of player 1 domination of player 3 domination of player 2

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𝝆

𝑻𝟐 𝑻𝟑 𝑻𝟒

Basin of attraction of an equilibrium π ,

with 𝐸 𝜌 = 𝟑, in a 3 player game.

A 2 dimensional cross in a 3 dimension strategy space. domination of player 1 domination of player 3 domination of player 2 defections of Pl’s 2 and 3 Pl’s 1 and 3 defections of Pl’s 1 and 2 defections of

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𝑜 = 200𝑁, each may join or not. Non-joiner payoff = 0. 𝑬 𝒇𝑲 = 𝟐. Equil: everybody joins 𝒇𝑲. Another unsustainable ex: All or Lose Joiner Payoff = 1 iff all 200𝑁 players join,

• 1 otherwise.