Algorithmic Game Theory and Economics: A very short...
Transcript of Algorithmic Game Theory and Economics: A very short...
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Algorithmic Game Theory and Economics: A very short introduction
Mysore Park Workshop
August, 2012
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PRELIMINARIES
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Game
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Rock Paper Scissors
0,0 -1,1 1,-1
1,-1 0,0 -1,1
-1,1 1,-1 0,0
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Bimatrix Notation
A B
Stra
tegi
es o
f P
laye
r A
Strategies of Player B
Stra
tegi
es o
f P
laye
r A
Strategies of Player B
Zero Sum Game: A + B = all zero matrix
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The Notion of Equilibrium
i i
j j
A B 𝐴𝑖𝑗 ≥ 𝐴𝑖′𝑗 ∀𝑖′ 𝐵𝑖𝑗 ≥ 𝐵𝑖𝑗′ ∀𝑗
′
i is a best response to j
j is a best response to i
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It needn’t exist
0,0 -1,1 1,-1
1,-1 0,0 -1,1
-1,1 1,-1 0,0
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Mixed Strategies and Equilibrium 𝑝1,𝑝
2,⋯
,𝑝𝑛
𝑝1,𝑝
2,⋯
,𝑝𝑛
𝑞1, 𝑞2, ⋯ , 𝑞𝑛 𝑞1, 𝑞2, ⋯ , 𝑞𝑛
A B
Expected utility of A = 𝑝𝑖𝐴𝑖𝑗𝑞𝑗𝑖,𝑗 = 𝑝⊤𝐴𝑞
Expected utility of B = 𝑝𝑖𝐵𝑖𝑗𝑞𝑗𝑖,𝑗 = 𝑝⊤𝐵𝑞
𝑝⊤𝐴𝑞 ≥ 𝑝′⊤𝐴𝑞 ∀𝑝′ 𝑝⊤𝐵𝑞 ≥ 𝑝⊤𝐵𝑞′ ∀𝑞′
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J. von Nuemann’s Minimax Theorem
Theorem (1928) If A+B=0, then equilibrium mixed strategies (p,q) exist.
max𝑝
min𝑞
𝑝⊤𝐴𝑞 = min𝑞
max𝑝
𝑝⊤𝐴𝑞
"As far as I can see, there could be no theory of games … without that theorem … I thought there was nothing worth publishing until the Minimax Theorem was proved"
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John Nash and Nash Equilibrium
Theorem (1950) Mixed Equilibrium always exist for finite games.
∃ 𝑝 , 𝑞 :
𝑝⊤𝐴𝑞 ≥ 𝑝′⊤𝐴𝑞 ∀𝑝′
𝑝⊤𝐵𝑞 ≥ 𝑝⊤𝐵𝑞′ ∀𝑞′
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Markets
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Walrasian Model: Exchange Economies
• N goods {1,2,…,N} • M agents {1,2,…,M}
• 𝑒𝑖 ≔ (𝑒𝑖1, 𝑒𝑖2, … , 𝑒𝑖𝑁) • 𝑈𝑖 𝒙 = 𝑈𝑖(𝑥𝑖1, … , 𝑥𝑖𝑁)
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Walrasian Model: Exchange Economies
• N goods {1,2,…,N} • M agents {1,2,…,M}
• 𝑒𝑖 ≔ (𝑒𝑖1, 𝑒𝑖2, … , 𝑒𝑖𝑁) • 𝑈𝑖 𝒙 = 𝑈𝑖(𝑥𝑖1, … , 𝑥𝑖𝑁)
• Prices 𝒑 = 𝑝1, … , 𝑝𝑁 ⇒ Demand (𝐷1, … , 𝐷𝑀)
𝐷𝑖 𝒑 = argmax { 𝑈𝑖(𝒙) ∶ 𝒑 ⋅ 𝒙 ≤ 𝒑 ⋅ 𝒆𝒊}
• 𝒑, 𝒙𝟏, … , 𝒙𝑴 are a Walrasian equilibrium if
o 𝒙𝒊 ∈ 𝐷𝑖 𝒑 for all 𝑖 ∈ [𝑀]
o 𝒙𝑗𝒊 = 𝑒𝑖𝑗𝒊𝑖 for all j ∈ [𝑁]
Utility Maximization
Market Clearing
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Leon Walras and the tatonnement
• Given price p, calculate demand for each good i.
• If demand exceeds supply, raise price.
• If demand is less than supply, decrease price.
Does this process converge? Does equilibrium exist?
1874
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Arrow and Debreu
Theorem (1954). If utilities of agents are continuous, and strictly quasiconcave, then Walrasian equilibrium exists.
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How does one prove such theorems?
∃ 𝑝 , 𝑞 : 𝑝⊤𝐴𝑞 ≥ 𝑝′⊤𝐴𝑞 ∀𝑝′
𝑝⊤𝐵𝑞 ≥ 𝑝⊤𝐵𝑞′ ∀𝑞′ Nash.
Define: Φ 𝑝, 𝑞 = (𝑝′, 𝑞′)
• 𝑝′ is a best-response to 𝑞
• 𝑞′ is a best-response to 𝑝
• Mapping is continuous.
Brouwers Fixed Point Theorem. Every continuous mapping from a convex, compact set to itself has a fixed point.
𝜙 𝑝, 𝑞 = 𝑝, 𝑞 ⇒ Equilibrium.
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EQUILIBRIUM COMPUTATION
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Support Enumeration Algorithms
∃ 𝑝 , 𝑞 : 𝑝⊤𝐴𝑞 ≥ 𝑝′⊤𝐴𝑞 ∀𝑝′; 𝑝⊤𝐵𝑞 ≥ 𝑝⊤𝐵𝑞′ ∀𝑞′ Nash.
Suppose knew the support S and T of (p,q).
𝑝 , 𝑞 : 𝑒𝑖
⊤𝐴𝑞 ≥ 𝑒𝑖′⊤𝐴𝑞; ∀𝑖 ∈ 𝑆, ∀𝑖′
𝑝⊤𝐵𝑒𝑗 ≥ 𝑝⊤𝐵𝑒𝑗′ ∀𝑗 ∈ 𝑇, ∀𝑗′
𝑝𝑖 = 0; 𝑞𝑗 = 0 ∀𝑖 ∉ S, ∀𝑗 ∉ 𝑇
Every point above is a Nash equilibrium.
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Lipton-Markakis-Mehta
Lemma. There exists 𝜖-approximate NE with
support size at most 𝐾 = 𝑂log 𝑛
𝜖2.
𝑝 , 𝑞 : 𝑝⊤𝐴𝑞 ≥ 𝑝′⊤𝐴𝑞 − 𝜖 ∀𝑝′;
𝑝⊤𝐵𝑞 ≥ 𝑝⊤𝐵𝑞′ − 𝜖 ∀𝑞′
𝝐-Nash
Given true NE (p,q), sample P,Q i.i.d. K times. Uniform distribution over P,Q is 𝜖-Nash.
Chernoff Bounds
Values in [0,1]
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Approximate Nash Equilibrium
Theorem (2003) [LMM] For any bimatrix game with at most 𝑛 strategies, an 𝜖-approximate Nash equilibrium can be
computed in time 𝑛𝑂 log 𝑛 /𝜖2
Theorem (2009) [Daskalakis, Papadimitriou] Any oblivious algorithm for Nash equilibrium runs
in expected time Ω(𝑛[0.8−0.34𝜖]log 𝑛).
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Lemke-Howson 1964
𝐴𝑧 ≤ 𝟏 𝑧 ≥ 0
∀𝑖: 𝑧𝑖 = 0 OR 𝐴𝑧 𝑖 = 1
• Nonzero solution ≈ Nash equilibrium.
• Every vertex has one “escape” and “entry”
• There must be a sink ≡ Non zero solution.
123
113
133
123
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How many steps?
• n strategies imply at most 𝑂(2𝑛) steps.
• Theorem (2004).[Savani, von Stengel] Lemke-Howson can take Ω 𝑐𝑛 steps, for some 𝑐 > 1.
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Hardness of NASH
Theorem (2005) [Daskalakis, Papadimitriou, Goldberg]
Computing Nash equilibrium in a 4-player game is PPAD hard.
Theorem (2006) [Chen, Deng]
Computing Nash equilibrium in a 2-player game is PPAD hard.
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I’ve heard of NP. What’s this PPAD?
• Subclass of “search” problems whose solutions are guaranteed to exist (more formally, TFNP)
• PPAD captures problems where existence is proved via a parity argument in a digraph.
• End of Line. Given G = ( 0,1 𝑛, 𝐴), in/out-deg ≤ 1 out-deg(0𝑛)=1, oracle for in/out(v); find v with in-deg(v)=1 and out-deg(v) = 0.
• PPAD is problems reducible to EOL.
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Summary of Nash equilibria
• Exact calculation is PPAD-hard. Even getting an FPTAS is PPAD-hard (Chen,Deng,Teng ‘07) Quasi-PTAS exists.
• Some special cases of games have had more success – Rank 1 games. (Ruta’s talk!)
– PTAS for constant rank (Kannan, Theobald ’07) sparse games (D+P ‘09) small probability games (D+P ’09)
• OPEN: Is there a PTAS to compute NE?
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General Equilibrium Story
• Computing General Equilibria is PPAD-hard. (Reduction to Bimatrix games)
• Linear case and generalizations can be solved via convex programming techniques. Combinatorial techniques.
• “Substitutability” makes tatonnement work.
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MECHANISM DESIGN
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Traditional Algorithms
ALGORITHM INPUT OUTPUT
MAXIMIZER N numbers (𝑣1, … , 𝑣𝑁)
𝑣max
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Auctions
𝑣1
𝑣2
𝑣𝑁
𝑣max
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Vickrey Auction • Setting
– Solicit bids from agents.
– Allocate item to one agent.
– Charge an agent (no more than bid.)
• Second Price Auction
– Assign item to the highest bidder.
– Charge the bid of the second highest bidder.
Theorem (1961). No player has an incentive to misrepresent bids in the second price auction.
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Mechanism Design Framework
• Feasible Solution Space. 𝐹 ⊆ 𝑹𝑁
• Agents. Valuations 𝑣𝑖: 𝐹 ↦ 𝑹 Reports type/bid 𝑏𝑖: 𝐹 ↦ 𝑹
• Mechanism.
– Allocation: 𝑥1, 𝑥2, … , 𝑥𝑁 ∈ 𝐹
– Prices: (𝑝1, 𝑝2, … 𝑝𝑁)
– Individual Rationality: 𝑝𝑖 ≤ 𝑏𝑖(𝑥𝑖)
– Incentive Compatibility: 𝑣𝑖 𝑥𝑖(𝑣𝑖) − 𝑝 𝑣𝑖 ≥ 𝑣𝑖 𝑥𝑖(𝑏𝑖) − 𝑝𝑖 𝑏𝑖
• Goal.
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Welfare Maximization: VCG
Goal. Maximize 𝑣𝑖 𝑥𝑖 : 𝑥1, … , 𝑥𝑁 ∈ 𝐹𝑖
VCG Mechanism
• Find 𝒙∗ which maximizes welfare.
• For agent 𝑎 calculate
o 𝒙 maximizing 𝑣𝑖 𝑥𝑖 : 𝒙 ∈ 𝐹 𝑖≠𝑎
o Charge 𝑝𝑎 = 𝑣𝑖 𝑥 𝑖 − 𝑣𝑖 𝑥𝑖∗
𝑖≠𝑎𝑖≠𝑎
• For single item ↦ second price auction.
o Utility = 𝑣𝑖 𝑥𝑖∗ − 𝑣𝑖(𝑥 𝑖)𝑖≠𝑎 𝑖
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Combinatorial Auctions
• N agents, M indivisible items. Utilities 𝑈𝑖.
• Hierarchies of utilities: subadditive, submodular, budgeted additive, ….
• Problem with VCG: maximization NP hard.
• Approximation and VCG don’t mix: even a PTAS doesn’t imply truthfulness.
• Algorithmic Mechanism Design Nisan, Ronen 1999.
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Minimax Objective: Load balancing
Goal. minimizemaxi
𝑣𝑖 𝑥𝑖 : 𝑥1, … , 𝑥𝑁 ∈ 𝐹
Age
nts
• VCG only works for SUM
• 𝑂(1)-approximation known for only special cases.
• General case = Ω(𝑛)?
• Characterization (?)
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Single Dimensional Settings
• Each agent controls one parameter privately.
• Monotone allocation algorithm ⇒ Truthful. (𝑥𝑖 should increase with 𝑣𝑖)
• Is there a setting where monotonicity constraint degrades performance?
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Techniques in AMD
• Maximal in Range Algorithms. Restrict range a priori s.t. maximization in P. Argue restriction doesn’t hurt much. Nisan-Ronen 99, Dobzinski-Nisan-Schapira ‘05, Dobzinski-Nisan ‘07, Buchfuhrer et al 2010….
• Randomized mechanisms. Linear programming techniques Lavi-Swamy 2005.
Maximal in Distributional Range Dobzinski-Dughmi 2009.
Convex programming Dughmi-Roughgarden-Yan 2011.
• Lower Bounds. Via characterizations. Dobzinski-Nisan 2007
Communication complexity. Nisan Segal 2001
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Summary of Mechanism Design
• AMD opens up a whole suite of algorithm questions.
• Lower bounds to performance. Are we asking the right question?
• Can characterizations developed by economists exploited algorithmically?
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WHAT I COULDN’T COVER
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THANK YOU