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
PRELIMINARIES
Game
Rock Paper Scissors
0,0 -1,1 1,-1
1,-1 0,0 -1,1
-1,1 1,-1 0,0
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
The Notion of Equilibrium
i i
j j
A B ๐ด๐๐ โฅ ๐ด๐โฒ๐ โ๐โฒ ๐ต๐๐ โฅ ๐ต๐๐โฒ โ๐
โฒ
i is a best response to j
j is a best response to i
It neednโt exist
0,0 -1,1 1,-1
1,-1 0,0 -1,1
-1,1 1,-1 0,0
Mixed Strategies and Equilibrium ๐1,๐
2,โฏ
,๐๐
๐1,๐
2,โฏ
,๐๐
๐1, ๐2, โฏ , ๐๐ ๐1, ๐2, โฏ , ๐๐
A B
Expected utility of A = ๐๐๐ด๐๐๐๐๐,๐ = ๐โค๐ด๐
Expected utility of B = ๐๐๐ต๐๐๐๐๐,๐ = ๐โค๐ต๐
๐โค๐ด๐ โฅ ๐โฒโค๐ด๐ โ๐โฒ ๐โค๐ต๐ โฅ ๐โค๐ต๐โฒ โ๐โฒ
J. von Nuemannโs Minimax Theorem
Theorem (1928) If A+B=0, then equilibrium mixed strategies (p,q) exist.
max๐
min๐
๐โค๐ด๐ = min๐
max๐
๐โค๐ด๐
"As far as I can see, there could be no theory of games โฆ without that theorem โฆ I thought there was nothing worth publishing until the Minimax Theorem was proved"
John Nash and Nash Equilibrium
Theorem (1950) Mixed Equilibrium always exist for finite games.
โ ๐ , ๐ :
๐โค๐ด๐ โฅ ๐โฒโค๐ด๐ โ๐โฒ
๐โค๐ต๐ โฅ ๐โค๐ต๐โฒ โ๐โฒ
Markets
Walrasian Model: Exchange Economies
โข N goods {1,2,โฆ,N} โข M agents {1,2,โฆ,M}
โข ๐๐ โ (๐๐1, ๐๐2, โฆ , ๐๐๐) โข ๐๐ ๐ = ๐๐(๐ฅ๐1, โฆ , ๐ฅ๐๐)
Walrasian Model: Exchange Economies
โข N goods {1,2,โฆ,N} โข M agents {1,2,โฆ,M}
โข ๐๐ โ (๐๐1, ๐๐2, โฆ , ๐๐๐) โข ๐๐ ๐ = ๐๐(๐ฅ๐1, โฆ , ๐ฅ๐๐)
โข Prices ๐ = ๐1, โฆ , ๐๐ โ Demand (๐ท1, โฆ , ๐ท๐)
๐ท๐ ๐ = argmax { ๐๐(๐) โถ ๐ โ ๐ โค ๐ โ ๐๐}
โข ๐, ๐๐, โฆ , ๐๐ด are a Walrasian equilibrium if
o ๐๐ โ ๐ท๐ ๐ for all ๐ โ [๐]
o ๐๐๐ = ๐๐๐๐๐ for all j โ [๐]
Utility Maximization
Market Clearing
Leon Walras and the tatonnement
โข Given price p, calculate demand for each good i.
โข If demand exceeds supply, raise price.
โข If demand is less than supply, decrease price.
Does this process converge? Does equilibrium exist?
1874
Arrow and Debreu
Theorem (1954). If utilities of agents are continuous, and strictly quasiconcave, then Walrasian equilibrium exists.
How does one prove such theorems?
โ ๐ , ๐ : ๐โค๐ด๐ โฅ ๐โฒโค๐ด๐ โ๐โฒ
๐โค๐ต๐ โฅ ๐โค๐ต๐โฒ โ๐โฒ Nash.
Define: ฮฆ ๐, ๐ = (๐โฒ, ๐โฒ)
โข ๐โฒ is a best-response to ๐
โข ๐โฒ is a best-response to ๐
โข Mapping is continuous.
Brouwers Fixed Point Theorem. Every continuous mapping from a convex, compact set to itself has a fixed point.
๐ ๐, ๐ = ๐, ๐ โ Equilibrium.
EQUILIBRIUM COMPUTATION
Support Enumeration Algorithms
โ ๐ , ๐ : ๐โค๐ด๐ โฅ ๐โฒโค๐ด๐ โ๐โฒ; ๐โค๐ต๐ โฅ ๐โค๐ต๐โฒ โ๐โฒ Nash.
Suppose knew the support S and T of (p,q).
๐ , ๐ : ๐๐
โค๐ด๐ โฅ ๐๐โฒโค๐ด๐; โ๐ โ ๐, โ๐โฒ
๐โค๐ต๐๐ โฅ ๐โค๐ต๐๐โฒ โ๐ โ ๐, โ๐โฒ
๐๐ = 0; ๐๐ = 0 โ๐ โ S, โ๐ โ ๐
Every point above is a Nash equilibrium.
Lipton-Markakis-Mehta
Lemma. There exists ๐-approximate NE with
support size at most ๐พ = ๐log ๐
๐2.
๐ , ๐ : ๐โค๐ด๐ โฅ ๐โฒโค๐ด๐ โ ๐ โ๐โฒ;
๐โค๐ต๐ โฅ ๐โค๐ต๐โฒ โ ๐ โ๐โฒ
๐-Nash
Given true NE (p,q), sample P,Q i.i.d. K times. Uniform distribution over P,Q is ๐-Nash.
Chernoff Bounds
Values in [0,1]
Approximate Nash Equilibrium
Theorem (2003) [LMM] For any bimatrix game with at most ๐ strategies, an ๐-approximate Nash equilibrium can be
computed in time ๐๐ log ๐ /๐2
Theorem (2009) [Daskalakis, Papadimitriou] Any oblivious algorithm for Nash equilibrium runs
in expected time ฮฉ(๐[0.8โ0.34๐]log ๐).
Lemke-Howson 1964
๐ด๐ง โค ๐ ๐ง โฅ 0
โ๐: ๐ง๐ = 0 OR ๐ด๐ง ๐ = 1
โข Nonzero solution โ Nash equilibrium.
โข Every vertex has one โescapeโ and โentryโ
โข There must be a sink โก Non zero solution.
123
113
133
123
How many steps?
โข n strategies imply at most ๐(2๐) steps.
โข Theorem (2004).[Savani, von Stengel] Lemke-Howson can take ฮฉ ๐๐ steps, for some ๐ > 1.
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.
Iโve heard of NP. Whatโs this PPAD?
โข Subclass of โsearchโ problems whose solutions are guaranteed to exist (more formally, TFNP)
โข PPAD captures problems where existence is proved via a parity argument in a digraph.
โข End of Line. Given G = ( 0,1 ๐, ๐ด), in/out-deg โค 1 out-deg(0๐)=1, oracle for in/out(v); find v with in-deg(v)=1 and out-deg(v) = 0.
โข PPAD is problems reducible to EOL.
Summary of Nash equilibria
โข Exact calculation is PPAD-hard. Even getting an FPTAS is PPAD-hard (Chen,Deng,Teng โ07) Quasi-PTAS exists.
โข Some special cases of games have had more success โ Rank 1 games. (Rutaโs talk!)
โ PTAS for constant rank (Kannan, Theobald โ07) sparse games (D+P โ09) small probability games (D+P โ09)
โข OPEN: Is there a PTAS to compute NE?
General Equilibrium Story
โข Computing General Equilibria is PPAD-hard. (Reduction to Bimatrix games)
โข Linear case and generalizations can be solved via convex programming techniques. Combinatorial techniques.
โข โSubstitutabilityโ makes tatonnement work.
MECHANISM DESIGN
Traditional Algorithms
ALGORITHM INPUT OUTPUT
MAXIMIZER N numbers (๐ฃ1, โฆ , ๐ฃ๐)
๐ฃmax
Auctions
๐ฃ1
๐ฃ2
๐ฃ๐
๐ฃmax
Vickrey Auction โข Setting
โ Solicit bids from agents.
โ Allocate item to one agent.
โ Charge an agent (no more than bid.)
โข Second Price Auction
โ Assign item to the highest bidder.
โ Charge the bid of the second highest bidder.
Theorem (1961). No player has an incentive to misrepresent bids in the second price auction.
Mechanism Design Framework
โข Feasible Solution Space. ๐น โ ๐น๐
โข Agents. Valuations ๐ฃ๐: ๐น โฆ ๐น Reports type/bid ๐๐: ๐น โฆ ๐น
โข Mechanism.
โ Allocation: ๐ฅ1, ๐ฅ2, โฆ , ๐ฅ๐ โ ๐น
โ Prices: (๐1, ๐2, โฆ ๐๐)
โ Individual Rationality: ๐๐ โค ๐๐(๐ฅ๐)
โ Incentive Compatibility: ๐ฃ๐ ๐ฅ๐(๐ฃ๐) โ ๐ ๐ฃ๐ โฅ ๐ฃ๐ ๐ฅ๐(๐๐) โ ๐๐ ๐๐
โข Goal.
Welfare Maximization: VCG
Goal. Maximize ๐ฃ๐ ๐ฅ๐ : ๐ฅ1, โฆ , ๐ฅ๐ โ ๐น๐
VCG Mechanism
โข Find ๐โ which maximizes welfare.
โข For agent ๐ calculate
o ๐ maximizing ๐ฃ๐ ๐ฅ๐ : ๐ โ ๐น ๐โ ๐
o Charge ๐๐ = ๐ฃ๐ ๐ฅ ๐ โ ๐ฃ๐ ๐ฅ๐โ
๐โ ๐๐โ ๐
โข For single item โฆ second price auction.
o Utility = ๐ฃ๐ ๐ฅ๐โ โ ๐ฃ๐(๐ฅ ๐)๐โ ๐ ๐
Combinatorial Auctions
โข N agents, M indivisible items. Utilities ๐๐.
โข Hierarchies of utilities: subadditive, submodular, budgeted additive, โฆ.
โข Problem with VCG: maximization NP hard.
โข Approximation and VCG donโt mix: even a PTAS doesnโt imply truthfulness.
โข Algorithmic Mechanism Design Nisan, Ronen 1999.
Minimax Objective: Load balancing
Goal. minimizemaxi
๐ฃ๐ ๐ฅ๐ : ๐ฅ1, โฆ , ๐ฅ๐ โ ๐น
Age
nts
โข VCG only works for SUM
โข ๐(1)-approximation known for only special cases.
โข General case = ฮฉ(๐)?
โข Characterization (?)
Single Dimensional Settings
โข Each agent controls one parameter privately.
โข Monotone allocation algorithm โ Truthful. (๐ฅ๐ should increase with ๐ฃ๐)
โข Is there a setting where monotonicity constraint degrades performance?
Techniques in AMD
โข Maximal in Range Algorithms. Restrict range a priori s.t. maximization in P. Argue restriction doesnโt hurt much. Nisan-Ronen 99, Dobzinski-Nisan-Schapira โ05, Dobzinski-Nisan โ07, Buchfuhrer et al 2010โฆ.
โข Randomized mechanisms. Linear programming techniques Lavi-Swamy 2005.
Maximal in Distributional Range Dobzinski-Dughmi 2009.
Convex programming Dughmi-Roughgarden-Yan 2011.
โข Lower Bounds. Via characterizations. Dobzinski-Nisan 2007
Communication complexity. Nisan Segal 2001
Summary of Mechanism Design
โข AMD opens up a whole suite of algorithm questions.
โข Lower bounds to performance. Are we asking the right question?
โข Can characterizations developed by economists exploited algorithmically?
WHAT I COULDNโT COVER
THANK YOU
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