Nash Equilibrium in Tullock Contests Aidas Masiliunas 1 1 - - PowerPoint PPT Presentation

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Nash Equilibrium in Tullock Contests

Aidas Masiliunas1

1Aix-Marseille School of Economics

Controversies in Game Theory III, ETH Zurich 2 June, 2016

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Rent-seeking (Tullock) contest

Two players compete for a prize (16 ECU) by making costly investments (x1, x2 ≤ 16) Higher investments increase the probability to win the prize Probability that player i receives the prize:

xi xi+xj

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Rent-seeking (Tullock) contest

Two players compete for a prize (16 ECU) by making costly investments (x1, x2 ≤ 16) Higher investments increase the probability to win the prize Probability that player i receives the prize:

xi xi+xj

Applications:

Competition for monopoly rents Investments in R&D Competition for a promotion/bonus Political contests

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Theory

E(π) =

xi xi+xj · 16 + 16 − xi

BRi(xj) : x∗

i = 16xj − xj

RNNE : x∗

i = 4, dominance solvable in three steps.

5 10 15 5 10 15

Standard preferences

Other plays Best Response 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 1 2 3 4 5 6 7 8 9 10 12 14 16

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Explanatory power of Nash equilibrium in experiments

7.04% of choices are exactly Nash 60.19% of choices are strictly dominated Investments are spread across the whole strategy space Experience does not help Less stability compared to auctions

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Comparative statics of Nash equilibrium

An alternative to point predictions is comparative statics Is behaviour sensitive to changes in the Nash prediction?

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Comparative statics of Nash equilibrium

An alternative to point predictions is comparative statics Is behaviour sensitive to changes in the Nash prediction? Players Nash Mean investment 2 250 325 3 222 283 4 188 302 5 160 322 9 99 326

Source: Lim, Matros & Turocy, 2014

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Why should players choose Nash equilibrium?

Interpretation #1: Nash equilibrium is the unique action profile that can be justified by common knowledge of rationality. Rationality = maximization of expected payoff given some belief.

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Rationalizable strategies

xi 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 BR(xi) 3 4 4 4 4 4 4 3 3 3 2 2 1 1 1 1

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Rationalizable strategies

xi 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 BR(xi) 3 4 4 4 4 4 4 3 3 3 2 2 1 1 1 1

Rationality

Rationalizable: 3, 4, 2, 1

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Rationalizable strategies

xi 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 BR(xi) 3 4 4 4 4 4 4 3 3 3 2 2 1 1 1 1 BR(BR(xi)) 4 4 4 4 4 4 4 4 4 4 4 4 3 3 3 3

Rationality

Rationalizable: 3, 4, 2, 1

Rationality + belief that the opponent is rational

Rationalizable: 3, 4

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Rationalizable strategies

xi 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 BR(xi) 3 4 4 4 4 4 4 3 3 3 2 2 1 1 1 1 BR(BR(xi)) 4 4 4 4 4 4 4 4 4 4 4 4 3 3 3 3 BR(BR(BR(xi))) 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4

Rationality

Rationalizable: 3, 4, 2, 1

Rationality + belief that the opponent is rational

Rationalizable: 3, 4

Rationality + belief that the opponent is rational + belief that the opponent believes in my rationality

Rationalizable: 4

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Rationalizable strategies

xi 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 BR(xi) 3 4 4 4 4 4 4 3 3 3 2 2 1 1 1 1 BR(BR(xi)) 4 4 4 4 4 4 4 4 4 4 4 4 3 3 3 3 BR(BR(BR(xi))) 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4

Rationality

Rationalizable: 3, 4, 2, 1

Rationality + belief that the opponent is rational

Rationalizable: 3, 4

Rationality + belief that the opponent is rational + belief that the opponent believes in my rationality

Rationalizable: 4

Epistemic definition of Nash equilibrium: common belief in rationality + simple belief hierarchy

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Why should players choose Nash equilibrium?

Nash equilibrium is the unique action profile that cannot be ruled out by common knowledge of rationality.

1

Players care about expected payoffs

2

Players have the ability to calculate expected payoffs and identify dominated strategies

3

Players believe that other players satisfy 1-2, and believe that they believe that they satisfy 1-2...

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Why should players choose Nash equilibrium?

Nash equilibrium is the unique action profile that cannot be ruled out by common knowledge of rationality.

1

Players care about expected payoffs

2

Players have the ability to calculate expected payoffs and identify dominated strategies

3

Players believe that other players satisfy 1-2, and believe that they believe that they satisfy 1-2...

Nash equilibrium is the rest point of various learning dynamics

Belief-based learning, e.g. Cournot best-response, fictitious play

Assumption 3 is not necessary

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Why should players choose Nash equilibrium?

Nash equilibrium is the unique action profile that cannot be ruled out by common knowledge of rationality.

1

Players care about expected payoffs

2

Players have the ability to calculate expected payoffs and identify dominated strategies

3

Players believe that other players satisfy 1-2, and believe that they believe that they satisfy 1-2...

Nash equilibrium is the rest point of various learning dynamics

Belief-based learning, e.g. Cournot best-response, fictitious play

Assumption 3 is not necessary

Payoff-based learning, e.g. reinforcement learning

Players must be willing to explore, remember past payoffs, receive accurate feedback.

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Which assumptions are violated?

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Preference-based explanations: joy of winning

Participants receive non-monetary utility from winning (Parco et al, 2005, Sheremeta, 2011) or lose utility after losing (Delgado et al., 2008). Sheremeta (2011) elicits joy of winning by implementing a contest where prize has no value.

5 10 15 5 10 15

Joy of winning with w=3

Other plays Best Response 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 1 2 3 4 5 6 7 8 9 10 12 14 16 5 10 15 5 10 15

Joy of winning with w = 8

Other plays Best Response 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 1 2 3 4 5 6 7 8 9 10 12 14 16

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Preference-based explanations: risk preferences

CRRA untility function: u(πi) = π1−ρ

i

1−ρ

Risk aversion if ρ = 0.5, risk seeking if ρ = −0.5

5 10 15 5 10 15

Risk aversion

Other plays Best Response 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 1 2 3 4 5 6 7 8 9 10 12 14 16 5 10 15 5 10 15

Risk seeking

Other plays Best Response 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 1 2 3 4 5 6 7 8 9 10 12 14 16

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Preference-based explanations: social preferences

Fehr & Schmidt (1999) inequality aversion: u(πi, πj) = πi − α(πj − πi) if πi ≤ πj πi − β(πi − πj) if πi > πj

5 10 15 5 10 15

Fehr and Schmidt (1999) inequality aversion

Other plays Best Response 1 2 3 4 5 6 7 8 9 10 12 14 16 1 2 3 4 5 6 7 8 9 11 13 15 a=0, b=0 a=0.5, b=0 a=1, b=0

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All preferences from Sheremeta (2015)

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”Behavioral Variation in Tullock Contests”, joint with F. Mengel and Ph. Reiss

Deviations from NE could be a result of bounded rationality Players optimize given the feedback in previous rounds. Noisy feedback prevents players from discovering optimal actions

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”Behavioral Variation in Tullock Contests”, joint with F. Mengel and Ph. Reiss

Deviations from NE could be a result of bounded rationality Players optimize given the feedback in previous rounds. Noisy feedback prevents players from discovering optimal actions Research questions:

Can we identify whether deviations from NE are a result of bounded rationality or of preferences? Is behavioral variability lower and choices closer to theoretical predictions when feedback is more informative?

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How informative is the feedback that players observe?

Reinforcement learning converges to NE as t → ∞ In experiments players rely on small samples of experience

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How informative is the feedback that players observe?

Reinforcement learning converges to NE as t → ∞ In experiments players rely on small samples of experience Suppose that players always choose the action that yielded highest average payoff in the past.

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How informative is the feedback that players observe?

Reinforcement learning converges to NE as t → ∞ In experiments players rely on small samples of experience Suppose that players always choose the action that yielded highest average payoff in the past.

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Feedback depends on other’s choices and lottery outcomes

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Treatment 1: eliminate lottery allocation

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Treatment 2: eliminate variability of opponent’s choices

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Treatment 3: eliminate both

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How easy is it to learn in different treatments?

Estimate the likelihood that action 4 will yield a higher average payoff than action 6.

Π(4) > Π(6)

Memory length % of iterations 10 20 30 40 50 25 50 75 100

• Shared prize, fixed actions

Shared prize, changing actions Lottery, fixed actions Lottery, changing actions

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How easy is it to learn in different treatments?

Estimate the likelihood that action 4 will yield a higher average payoff than action 6.

Π(4) > Π(6)

Memory length % of iterations 10 20 30 40 50 25 50 75 100

• Shared prize, fixed actions

Shared prize, changing actions Lottery, fixed actions Lottery, changing actions

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How easy is it to learn in different treatments?

Estimate the likelihood that action 4 will yield a higher average payoff than action 6.

Π(4) > Π(6)

Memory length % of iterations 10 20 30 40 50 25 50 75 100

• Shared prize, fixed actions

Shared prize, changing actions Lottery, fixed actions Lottery, changing actions

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How easy is it to learn in different treatments?

Estimate the likelihood that action 4 will yield a higher average payoff than action 6.

Π(4) > Π(6)

Memory length % of iterations 10 20 30 40 50 25 50 75 100

• Shared prize, fixed actions

Shared prize, changing actions Lottery, fixed actions Lottery, changing actions

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Procedure

40 rounds, divided into 4 blocks of 10 rounds Each block divided into experimentation phase (rounds 1-5) and incentivized phase (rounds 6-10)

1 5 10 6 11 15 16 20 21 26 30 25 35 31 36 40

Non-incentivized Non-incentivized Non-incentivized Non-incentivized Incentivized Incentivized Incentivized Incentivized

Block 4 Block 3 Block 2 Block 1

One round from each block randomly chosen for payment Incentivized numeracy test at the end of the experiment Average earnings 15.15 euro, duration 60 minutes

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Explanatory power of Nash equilibrium

Changing actions Fixed actions Lottery EV Lottery EV P(x = NE) 7.04% 13.33%

• P(x = BR)
• 22.50%

65.23% P(|x − NE| ≤ 1) 25.74% 32.78%

• P(|x − BR| ≤ 1)
• 47.95%

83.64% P(x > 4) 60.19% 62.78% 51.36% 16.14%

Absolute value of deviation from equilibrium significantly different between EV/Fixed treatment and the other three treatments, but not in

• ther comparisons.

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Behavioral variation

Is the distribution of choices more concentrated? (not necessarily around NE) Entropy measures the stochastic variation of a random variable (0 = one strategy always chosen, 4 = all strategies chosen with equal frequency): H = −

• i=1...16

pi log(pi)

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Behavioral variation

Is the distribution of choices more concentrated? (not necessarily around NE) Entropy measures the stochastic variation of a random variable (0 = one strategy always chosen, 4 = all strategies chosen with equal frequency): H = −

• i=1...16

pi log(pi)

Changing actions Fixed actions Lottery EV Lottery EV Entropy 3.22 2.79 2.45 1.50

• Std. Dev.

3.28 2.56 3.15 1.16

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Best-response curves in Fixed treatments

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Stability of choices and convergence

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Stability of choices and convergence

Changing strategies between rounds in experimentation and incentivized rounds.

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Replacing humans by computers

Playing against a computer player is different than playing against a human player: no social preferences, lower joy of winning (?)

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Replacing humans by computers

Playing against a computer player is different than playing against a human player: no social preferences, lower joy of winning (?) Additional treatment replacing computers by human players. All effects replicate if Fixed/EV treatment is replaced by this treatment.

Changing actions Fixed actions Lottery EV Lottery EV EV-Human P(x = NE) 7.04% 13.33%

• P(x = BR)
• 22.50%

65.23% 50.42% P(|x − NE| ≤ 1) 25.74% 32.78%

• P(|x − BR| ≤ 1)
• 47.95%

83.64% 74.58% P(x > 4) 60.19% 62.78% 51.36% 16.14% 23.33% Entropy 3.22 2.79 2.45 1.50 1.13

• Std. Dev.

3.28 2.56 3.15 1.16 0.91

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Strategic uncertainty vs stability

Matching players to computers has two effects:

The action of the other party is stable over time, hence it is easier to learn. Players face no strategic uncertainty, hence it is easier to

• ptimize

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Strategic uncertainty vs stability

Matching players to computers has two effects:

The action of the other party is stable over time, hence it is easier to learn. Players face no strategic uncertainty, hence it is easier to

• ptimize

Is stability of choices necessary in addition to the removal of strategic uncertainty? Design: computer plays actions from the baseline contest, players know these actions.

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Strategic uncertainty vs stability

Matching players to computers has two effects:

The action of the other party is stable over time, hence it is easier to learn. Players face no strategic uncertainty, hence it is easier to

• ptimize

Is stability of choices necessary in addition to the removal of strategic uncertainty? Design: computer plays actions from the baseline contest, players know these actions.

Changing actions Changing but known Fixed actions Lottery EV Lottery EV Lottery EV P(a = NE) 7.04% 13.33%

• P(a = BR)
• 7.59%

25.37% 22.50% 65.23% P(|a − NE| ≤ 1) 25.74% 32.78%

• P(|a − BR| ≤ 1)
• 25.00%

51.85% 47.95% 83.64% P(a > 4) 60.19% 62.78% 62.96% 47.04% 51.36% 16.14%

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Strategic uncertainty vs stability

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Contests with forgone payoff information

Conclusion from the first paper: when feedback is more informative about the quality of actions, players make better choices. Can we improve the quality of feedback without changing the nature of the game?

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Contests with forgone payoff information

Conclusion from the first paper: when feedback is more informative about the quality of actions, players make better choices. Can we improve the quality of feedback without changing the nature of the game? Hypothesis: more information and higher quality of information increases the rate of learning Design: 10 rounds of standard contest, 20 rounds of contest with foregone payoff information, 10 rounds of standard contest

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”Contests with foregone payoff information”

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Hypotheses: reinforcement learning simulation

Π(2) > Π(4)

Memory length % of iterations 10 20 30 40 50 25 50 75 100

• Same actions, same random numbers

Different actions, same random numbers Same actions, different random numbers Different actions, different random numbers

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Results: average investments

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Results: dominated strategies

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Payoff based learning, joint with H. Nax

Calculating expected values is very complicated Convergence is much higher when players can use a payoff table/calculator and with neutral framing

200 400 600 800 invest 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

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Summary

Nash equilibrium has a very low explanatory power in Tullock contests Explanatory power is much higher when actions have direct payoff consequences Providing additional feedback about foregone payoff information does not improve the explanatory power Paying the expected payoffs does not improve learning, unless players know these payoffs.