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INTRODUCTION TO GAME THEORY
Heinrich H. Nax Bary S. R. Pradelski&
[email protected] [email protected]
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Introduction
To begin: let’s play the following game
Rules:1 Players: All of you2 Actions: Choose a non-negative integer between 0 and 1003 Outcome: The player with the number closest to half the average of all
submitted numbers wins.4 Payoffs: He/she will will 20 CHF.5 In case of several winners, divide payment by number of winners and
pay all winners.6 Visit https://doodle.com/poll/knzzkxzx9w3sheqy once and leave a
guess with name or pseudonym by May 25, 2020
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Introduction
Keynesian (p = 1/2-) Beauty Contest (Moulin (1986))
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Introduction
What usually happens (p = 2/3)...
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Introduction
Acknowledgments
Peyton Young (Oxford, LSE)
Bernhard von Stengel (LSE)
Francoise Forges (Paris Dauphine)
Jeff Shamma (Georgia Tech, KAUST)
Joergen Weibull (Stockholm, TSE)
Andreas Diekmann (ETHZ)
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Introduction
Game theory
A tour of its people, applications and concepts1 von Neumann2 Nash3 Aumann, Schelling, Selten, Shapley4 Today
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Introduction
John von Neumann (1903-1957); polymath; ETH 1923–1926https://gametheory.online/johnny 7 / 41
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Introduction
What is game theory?
A mathematical language to express models of, as Myerson says:“conflict and cooperation between intelligent rational decision-makers”
In other words, interactive decision theory (Aumann)
Dates back to von Neumann & Morgenstern (1944)
Most important solution concept: the Nash (1950) equilibrium
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Introduction
Games and Non-Games
What is a game? And what is not a game?
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Introduction
Uses of game theory
Prescriptive agenda versus descriptive agenda“Reverse game theory”/mechanism design:
“in a design problem, the goal function is the main given, while themechanism is the unknown.” (Hurwicz)
The mechanism designer is a game designer. He studiesWhat agents would do in various gamesAnd what game leads to the outcomes that are most desirable
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Introduction
Game theory revolutionized several disciplines
Biology (evolution, conflict, etc.)
Social sciences (economics, sociology, political science, etc.)
Computer science (algorithms, control, etc.)
game theory is now applied widely (e.g. regulation, online auctions,distributed control, medical research, etc.)
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Introduction
Its impact in economics (evaluated by Nobel prizes)
1972: Ken Arrow − general equilibrium
1994: John Nash, Reinhard Selten, John Harsanyi − solution concepts
2005: Tom Schelling and Robert Aumann − evolutionary game theoryand common knowledge
2007: Leonid Hurwicz, Eric Maskin, Roger Myerson − mechanismdesign
2009: Lin Ostrom − economic governance, the commons
2012: Al Roth and Lloyd Shapley − market design
2014: Jean Tirole − markets and regulation
2016: Oliver Hart and Bengt Holmström − contract theory
2017: Richard Thaler − limited rationality, social preferences
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Introduction
Part 1: game theory
“Introduction” / Tour of game theory
Non-cooperative game theoryNo binding contracts can bewrittenPlayers are individualsMain solution concepts:
Nash equilibriumStrong equilibrium
Cooperative game theory
Binding contract can be written
Players are individuals andcoalitions of individualsMain solution concepts:
CoreShapley value
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Introduction
Noncooperative game theory
John Nash (1928-2015)
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Introduction
A noncooperative game (normal-form)
players: N = {1, 2, . . . , n} (finite)actions/strategies: (each player chooses si from his own finite strategyset; Si for each i ∈ N)
resulting strategy combination: s = (s1, . . . , sn) ∈ (Si)i∈N
payoffs: ui = ui(s)payoffs resulting from the outcome of the game determined by s
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Introduction
Some 2-player examples
Prisoner’s dilemma − social dilemma, tragedy of the commons,free-riding
Conflict between individual and collective incentivesHarmony − aligned incentives
No conflict between individual and collective incentivesBattle of the Sexes − coordination
Conflict and alignment of individual and collective incentivesHawk dove/Snowdrift − anti-coordination
Conflict and alignment of individual and collective incentivesMatching pennies − zero-sum, rock-paper-scissor
Conflict of individual incentives
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Introduction
Player 2Heads Tails
Player 1 Heads 1,-1 -1,1Tails -1,1 1,-1
Matching pennies
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Introduction
Confess Stay quietA A
Confess -6 -10B -6 0
Stay quiet 0 -2B -10 -2
Prisoner’s dilemma
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Introduction
WOMANBoxing Shopping
MAN Boxing 2,1 0,0Shopping 0,0 1,2
Battle of the sexes
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Introduction
Player 2Hawk Dove
Player 1 Hawk -2,-2 4,0Dove 0,4 2,2
Hawk-Dove game
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Introduction
Company BCooperate Not Cooperate
Company A Cooperate 9,9 4,7Not Cooperate 7,4 3,3
Harmony game
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Introduction
Equilibrium
Equilibrium/solution concept:
An equilibrium/solution is a rule that maps the structure of a game intoan equilibrium set of strategies s∗.
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Introduction
Nash Equilibrium
Definition: Best-response
Player i’s best-response (or, reply) to the strategies s−i played by allothers is the strategy s∗i ∈ Si such that
ui(s∗i , s−i) > ui(s′i, s−i) ∀s′i ∈ Si and s′i 6= s∗i
Definition: (Pure-strategy) Nash equilibrium
All strategies are mutual best responses:
ui(s∗i , s−i) > ui(s′i, s−i) ∀s′i ∈ Si and s′i 6= s∗i
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Introduction
Confess Stay quietA A
Confess -6 -10B -6 0
Stay quiet 0 -2B -10 -2
Prisoner’s dilemma: both players confess (defect)
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Introduction
WOMANBoxing Shopping
MAN Boxing 2,1 0,0Shopping 0,0 1,2
Battle of the sexes: coordinate on either option
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Introduction
Player 2Heads Tails
Player 1 Heads 1,-1 -1,1Tails -1,1 1,-1
Matching pennies: none (in pure strategies)
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Introduction
Player 2Hawk Dove
Player 1 Hawk -2,-2 4,0Dove 0,4 2,2
Hawk-dove: either of the two hawk-dove outcomes
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Introduction
Company BCooperate Not Cooperate
Company A Cooperate 9,9 4,7Not Cooperate 7,4 3,3
Harmony: both cooperate
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Introduction
Pure-strategy N.E. for our 2-player examples
Prisoner’s dilemma − social dilemmaUnique NE − socially undesirable outcome
Harmony − aligned incentivesUnique NE − socially desirable outcome
Battle of the Sexes − coordinationTwo NE − both Pareto-optimal
Hawk dove/Snowdrift − anti-coordinationTwo NE − Pareto-optimal, but perhaps Dove-Dove “better”
Matching pennies − zero-sum, rock-paper-scissorNo (pure-strategy) NE
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Introduction
How about our initial game
Remember the rules were:1 Choose a number between 0 and 1002 The player with the number closest to half the average of all submitted
numbers wins
What is the Nash Equilibrium?
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Introduction
0
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Introduction
Iterative reasoning...
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Introduction
Braess’ Paradox
New road worsens congestion!
The story:
60 people travel from S to D
Initially, there is no middle road.The NE is such that 30 peopletravel one way, the others theother way, and each driver drives90 mins.
A middle road is build. Thisroad is super efficient. Noweveryone will use it and drivethe same route, and the NE willworsen to 119/120 mins.
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Introduction
Cooperative games
The Nash equilibrium may not coincide with the outcome that is collectivelypreferable. Can players “cooperate" to achieve such an outcome?
Suppose players can write binding agreements and directly transferutility– e.g.:
Contract 1: Player 1 plays ‘Hawk’, player 2 plays ‘Dove’. Of the totalpayoffs, 1 and 2 receive equal shares.
orContract 2: Both players play ‘Boxing’. Of the total payoffs, Man gets 1.6and Woman gets 1.4.
Then the value of the game in terms of a cooperative agreement is generallygreater than the sum of the payoffs from the Nash equilibrium.
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Introduction
Confess Stay quietA A
Confess -6 -10B -6 0
Stay quiet 0 -2B -10 -2
v(12) = −2− 2 = −4v(1) = v(2) = −6Cooperative value=v(12) > v(1) + v(2) =Nash equilibrium payoffs
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Introduction
WOMANBoxing Shopping
MAN Boxing 2,1 0,0Shopping 0,0 1,2
v(12) = 1 + 2 = 3v(1) = v(2) = 0Cooperative value=Nash equilibrium payoffs=v(12) > v(1) + v(2): payoffscan be split differently/more evenly
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Introduction
DawnHawk Dove
Gary Hawk -2,-2 4,0Dove 0,4 2,2
v(12) = 4 + 0 = 2 + 2 = 4v(1) = v(2) = −2Cooperative value=Nash equilibrium payoffs=v(12) > v(1) + v(2): payoffscan be split differently/more evenly, achievable by dove-dove
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Introduction
Company BCooperate Not Cooperate
Company A Cooperate 9,9 4,7Not Cooperate 7,4 3,3
v(12) = 9 + 9 = 18v(1) = v(2) = 3Cooperative value=Nash equilibrium payoffs=v(12) > v(1) + v(2), butpayoffs can be split differently/more evenly
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Introduction
Schedule (preliminary) I1) Introduction: a quick tour of game theory Heinrich Nax2) Cooperative game theory Heinrich Nax
•Core and Shapley value•Matching markets
3) Non-cooperative game theory: Normal form Bary Pradelski•Utilities•Best replies
4) The Nash equilibrium Bary Pradelski•Proof•Interpretations and refinements
5) Non-cooperative game theory: dynamics Bary Pradelski•Sub-game perfection and Bayes-Nash equilibrium•Repeated games
6) Game theory: evolution Bary Pradelski•Evolutionary game theory•Algorithms in computer science (Price of anarchy)
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Introduction
Schedule (preliminary) II7) Experimental game theory Heinrich Nax
•Observing human behavior/experiments•Behavioral game theory
8) Applications Heinrich Nax•Common pool resources•Distributed control
9) Bargaining Heinrich Nax•Solution concepts•Nash program
10) Auctions Bary Pradelski•English, Dutch, Sealed, Open•Equivalence and Real-world examples: 3G, Google, etc
11) John von Neumann lecture – Herve Moulin May 29, 202012) FEEDBACK Q&A Heinrich Nax
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Introduction
THANKS EVERYBODY
Keep checking the website for new materials as we progress:http://gametheory.online/project_show/9
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