algorithmic game theory - basic solution concepts and computational issues Éva tardos and vijay v....

Algorithmic Game Theory - Basic Solution Concepts and Computational Issues

Éva Tardos and Vijay V. Vazirani

Presentation

Reiknirit, rökfræði og reiknanleiki - 08.73.11 Anna Ólafsdóttir Björnsson

Overview

Introduction to basic game-theoretic definitions and tools

Different methods are suitable for different games

Pros and cons of some of the different methods introduced

2Anna Ólafsdóttir Björnsson

The questions the paper deals with

Game theory methods Games, Strategies, Cost and Payoffs Introducing basic solution concepts How easy is it to find an equilibrium Does "natural game play” lead the

players to an equilibrium? Examples (Nash and others)


Game theory methods

There are several ways of presenting game theory

For each game may be a different preference

Cost matrices show what each player gains/loses in each move

Equilibriums are know for several games


Example: Prisoner’s Dilemma

A classic and well studied game is the Prisoner’s Dilemma

2 prisoners can a) confess b) remain silent

Each can reduce own sentence by confessing but his/her confession is only rewarded if the other one chooses to remain silent


Cost matrix for Prisoner’s Dilemma

4

4

5

1

1

5

2

2

P2

P1

Confess Silent

Confess

Silent


Games, Strategies, Cost, Payoffs

Simultaneous Move Game: Each player i selects game strategy defined in

a vector of strategies si ∈ Si

Vector of strategies of all players: s = (s1 ... Sn)

Set of all possibities to pick strategy: S = xiSi

Assign values for each move for a player Utilities ui: S-R and Costs ci: S-R Costs and payoffs can be used

interchangeably since ui(s) = -ci(s)


Basic solution concepts

What affects the strategy chosen: Unique best strategy or not? Individual

choices – often low payoff Example: Prisoner’s Dilemma

Strategy chosen by one player effects the other – but individually chosen

Willingness to risk for a better payoff A 3rd party deciding for both players

Example: Traffic lights

Anna Ólafsdóttir Björnsson 8

Basic solution concepts (cont.)

Games that have unique best strategy – "Dominant strategy solution”

Prisoners Dilemma, Pollution Game etc. Solution that may not give optimal payoff

to any of the players Designing such games. Example: Auction

and Second price auction-Vickery auction Where the highest bidder pays 2nd highest

price and each player independant of others ... but "games rarely possess dominent

strategy solution”



More common than using unique best strategy:

Player tries to maximize the payoff Strategy where no single player can

individually improve his/her welfare Players cannot change their strategy Get better results not changing

With and without use of Nash Equilibria Unique best strategy is a Nash Equilibrium ...but not all Nash Equilibria are unique Tragedy of the commons, Battle of sexes



Any game with a finite set of players and finit set of strategies has a Nash equilibrium of mixed strategies


How easy to find an equilibrium?

Correlated equilibria – can be found in polynominal time

Two persons zero-sum lineral computing lead to Nash (PPAD)


Natural game play and equilibrium

Does "natural game play” lead the players to an equilibrium?

Assume that most natural "game playing” strategy is "best response”

Learning strategies can improve results Not all types of games lead to Nash

equilibria but instead to correlated equilibria if "learning strategies” are used


More types of games

Games with multiple turns of moves can be reduced to simultaneous move games and thus to Nash equilibria

Without full information - Bayesian Games – use probability distribution

Cooperative games – players coordinate their actions


Different types of cooperative games

Name Description Methods “Games”

Strong Nash Equilibrium

No subset A of players have a way to simultaneously change their strategies, improving each of the participants welfare

sA vector of players startegiess-A vector of strategies for players outside Aui(sA, s-A) but NOT

ui(s) ≤ ui(s’A, s-A)

Drawback - applies to few games:

Stable marriage problem

Fair division and cost-sharingTransferable utility games

Game dividing some value or sharing cost between participants fairly. Cost-sharing is in the core if no subset of players would decrease their shares by breaking away from the whole set

N = all playersA = subset of players associated with a costc(A), c(N), v(A) cost of serving playersCost share vector in core if:

∑i ∈Axi ≤ c(A)

Shapley value -Order player set N 1, ... N marginal cost of player i is c(Ni)- c(Ni-1)


Market “games” and their algorithimc issues

Mostly non-algorithmic theory - need to find (more) algorithms

Lemma: A uniform price of x on all goods is feasible iff all goods can be sold in such a way that each buyer gets goods that she is interested in

Lemma: The value x* is feasible for the problem restrict to goods in A-S* and buyers in B-Γ(S*). Furthermore, in the subgraph of G induced on A-S* and B-Γ(S*), all the vertices have nonzero degree.

There exists an algorithm that computes equilibrium prices and allocations in polynominal time (pg.25).


References

Éva Tardos and Vijay V. Vazirani: Basic Solution Concepts and Computational Issues. Algorithmic Game Theory

Algorithic Game Theory. Some other articles as a side material.

Wikipedia. Game Theory. (www.wikipedia.org)


algorithmic game theory - basic solution concepts and computational issues Éva tardos and vijay v....

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