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Understanding Rational Behavior and Nash Equilibria in Algorithmic Game Theory, Slides of Game Theory

Carnegie Mellon University (CMU)Game Theory

An introduction to Algorithmic Game Theory, a subfield of computer science that deals with systems of interacting agents, each with their own interests. the basics of Game Theory, including the concept of Nash Equilibrium and its application to various examples. It also touches upon minimax optimal strategies and their relationship to Nash Equilibria.

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2021/2022

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A Basic Introduction to Game
Theory
Avrim Blum
Game theory
ā€¢Field developed by economists to study social
& economic interactions.
ā€“Wanted to understand why people behave the
way they do in different economic situations.
Effects of incentives. Rational explanation of
behavior.
Game theory
ā€¢Field developed by economists to study social
& economic interactions.
ā€“Wanted to understand why people behave the
way they do in different economic situations.
Effects of incentives. Rational explanation of
behavior.
ā€¢ ā€œGameā€ = interaction between parties with
their own interests. Could be called
ā€œinteraction theoryā€.
ā€¢Big in CS for understanding large systems:
ā€“Internet routing, social networks, e-commerce
ā€“Problems like spam etc.
Led to new subfield: Algorithmic
Game Theory
Theory and algorithms for systems of interacting
agents, each with their own interests in mind.
Game Theory: Setting
ā€¢Have a collection of participants, or
players
.
ā€¢Each has a set of choices, or
strategies
for
how to play/behave.
ā€¢Combined behavior results in
payoffs
(satisfaction level) for each player.
Most examples today will involve just 2 players
(which will make them easier to picture, as
will become clear in a momentā€¦)
Example: walking on the sidewalk
ā€¢What side of sidewalk should I walk on?
ā€¢Two options for you (left or right). Same
for person walking towards you.
ā€¢Can describe payoffs in matrix:
(1,1) (-1,-1)
(-1,-1) (1,1)
Left
Right
Left Right
person
walking
towards you
you
Your payoff for RR His payoff for RR
street to drive on
pf3
pf4
pf5

Partial preview of the text

Download Understanding Rational Behavior and Nash Equilibria in Algorithmic Game Theory and more Slides Game Theory in PDF only on Docsity!

A Basic Introduction to Game

Theory

Avrim Blum

Game theory

  • Field developed by economists to study social

& economic interactions.

  • Wanted to understand why people behave the way they do in different economic situations. Effects of incentives. Rational explanation of behavior.

Game theory

  • Field developed by economists to study social

& economic interactions.

  • Wanted to understand why people behave the way they do in different economic situations. Effects of incentives. Rational explanation of behavior.
  • ā€œGameā€ = interaction between parties with

their own interests. Could be called

ā€œinteraction theoryā€.

  • Big in CS for understanding large systems:
    • Internet routing, social networks, e-commerce
    • Problems like spam etc.

Led to new subfield: Algorithmic

Game Theory

Theory and algorithms for systems of interacting agents, each with their own interests in mind.

Game Theory: Setting

  • Have a collection of participants, or players.
  • Each has a set of choices, or strategies for

how to play/behave.

  • Combined behavior results in payoffs

(satisfaction level) for each player.

Most examples today will involve just 2 players

(which will make them easier to picture, as

will become clear in a momentā€¦)

Example: walking on the sidewalk

  • What side of sidewalk should I walk on?
  • Two options for you (left or right). Same

for person walking towards you.

  • Can describe payoffs in matrix: (1,1) (-1,-1) (-1,-1) (1,1) Left Right Left Right person walking you towards you Your payoff for RR His payoff for RR

street to drive on

Could be randomized

Key notion

Left Right Left Right person walking you towards you Your payoff for RR His payoff for RR

  • ā€œNash Equilibriumā€: pair of strategies such

that each player is playing a best-response

to the other. Neither has an incentive to

change.

Example: prisonerā€™s dilemma

  • Consider two companies deciding whether to

install pollution controls.

  • Imagine pollution controls cost $4 but

improve everyoneā€™s environment by $

control donā€™t control control donā€™t control Need to add extra incentives to get good overall behavior.What do equilibria look like here? For both, defecting is dominant strategy

Example: matching pennies / penalty shot

  • Shooter can choose to shoot left or shoot right.
  • Goalie can choose to dive left or dive right.
  • If goalie guesses correctly, (s)he saves the day. If not, itā€™s a goooooaaaaall! Vice-versa for shooter. (0,0) (1,-1) (1,-1) (0,0) Left Right Left Right goalie shooter

Each playing 50/50 is a No deterministic equilibrium Nash equilibrium

GOAALLL!!!

Nash (1950)

  • Proved that if you allow randomized (mixed)

strategies then every game has at least one

equilibrium.

  • I.e., a pair of (randomized) strategies that is

stable in the sense that each is a best

response to the other in terms of expected

payoff.

  • For this, and its implications, Nash received

the Nobel prize.

Game theory terminology

  • Rows and columns called pure strategies.
  • Randomized algs called mixed strategies.
  • Often describe in terms of 2 matrices R, C.

(p,q) is Nash equilib if pTRq Āø eiTRq 8 i and

pTCq Āø pTCej 8 j.

R

C

Basic facts

  • (p,q) is NashEq if pTRq Āø eiTRq 8 i, pTCq Āø pTCej 8 j.
  • ) for all i s.t. pi > 0 we have eiTRq = maxiā€™ eiā€™TRq
  • ) for all j s.t. qj > 0 we have pTCej = maxjā€™ pTCejā€™ 1 - 1
  • 1 1

R

C

Minimax-optimal strategies

  • Can solve for minimax optimal strategy using Linear Programming: Variables p , v. Maximize v subject to: - p Ā¢ Mj Āø v, for all j. - p is legal prob dist (pi Āø 0, ļƒ„i pi = 1). shooter goalie (Ā½,-Ā½) (1,-1) (1,-1) (0,0) Left Right Left Right GOAALLL!!! 50/

Minimax Theorem (von Neumann 1928)

  • Every 2-player zero-sum game has a unique

value V.

  • Minimax optimal strategy for R guarantees

Rā€™s expected gain at least V.

  • Minimax optimal strategy for C guarantees

Cā€™s expected loss at most V.

Counterintuitive: Means it doesnā€™t hurt to publish your strategy if both players are optimal. (Borel had proved for symmetric 5x5 but thought was false for larger games)

Nash ) Minimax

  • Nashā€™s theorem actually gives minimax thm

as a corollary.

  • Pick some NE and let V = value to row player in that equilibrium.
  • Since itā€™s a NE, neither player can do better even knowing the (randomized) strategy their opponent is playing.
  • So, theyā€™re each playing minimax optimal.

Nash ) Minimax

  • On the other hand, for minimax, also have

very constructive, algorithmic arguments:

  • Can solve for minimax optimum using linear programming in time poly(n) (n = size of game)
  • Have adaptive procedures that in repeated play guarantee to approach/beat best fixed strategy in hindsight
  • But for Nash, no efficient procedures to

find: NP-hard to find equilib with special

properties, PPAD-hard just to find one.

Can use notion of minimax

optimality to explain bluffing

in poker

Simplified Poker (Kuhn 1950)

  • Two players A and B.
  • Deck of 3 cards: 1 , 2 , 3.
  • Players ante $1.
  • Each player gets one card.
  • A goes first. Can bet $1 or pass.
    • If A bets, B can call or fold.
    • If A passes, B can bet $1 or pass.
      • If B bets, A can call or fold.
  • High card wins (if no folding). Max pot $2.
  • Two players A and B. 3 cards: 1 , 2 , 3.
  • Players ante $1. Each player gets one card.
  • A goes first. Can bet $1 or pass.
    • If A bets, B can call or fold.
    • If A passes, B can bet $1 or pass.
      • If B bets, A can call or fold.

Writing as a Matrix Game

  • For a given card, A can decide to
    • Pass but fold if B bets. [PassFold]
    • Pass but call if B bets. [PassCall]
    • Bet. [Bet]
  • Similar set of choices for B.

Can look at all strategies as a

big matrixā€¦

[FP,FP,CB] [FP,CP,CB] [FB,FP,CB] [FB,CP,CB]

[PF,PF,PC]

[PF,PF,B]

[PF,PC,PC]

[PF,PC,B]

[B,PF,PC]

[B,PF,B]

[B,PC,PC]

[B,PC,B]

And the minimax optimal

ā€¢ A:^ strategies areā€¦

  • If hold 1, then 5/6 PassFold and 1/6 Bet.
  • If hold 2, then Ā½ PassFold and Ā½ PassCall.
  • If hold 3, then Ā½ PassCall and Ā½ Bet.

Has both bluffing and underbiddingā€¦

  • B:
    • If hold 1, then 2/3 FoldPass and 1/3 FoldBet.
    • If hold 2, then 2/3 FoldPass and 1/3 CallPass.
    • If hold 3, then CallBet Minimax value of game is ā€“ 1/18 to A.

How to prove existence of NE

  • Proof will be non-constructive.
  • Notation:
    • Assume an nxn matrix.
    • Use (p 1 ,...,pn) to denote mixed strategy for row player, and (q 1 ,...,qn) to denote mixed strategy for column player.

Proof

  • Weā€™ll start with Brouwerā€™s fixed point

theorem.

  • Let S be a bounded convex region in Rn^ and let f:S! S be a continuous function.
  • Then there must exist x 2 S such that f(x)=x.
  • x is called a ā€œfixed pointā€ of f.
  • Simple case: S is the interval [0,1].
  • We will care about:
  • S = {(p,q): p,q are legal probability distributions on 1,...,n}. I.e., S = simplexn Ā£ simplexn

Proof (cont)

  • S = {(p,q): p,q are mixed strategies}.
  • Want to define f(p,q) = (pā€™,qā€™) such that:
    • f is continuous. This means that changing p or q a little bit shouldnā€™t cause pā€™ or qā€™ to change a lot.
    • Any fixed point of f is a Nash Equilibrium.
  • Then Brouwer will imply existence of NE.

Algorithmic Game Theory

Algorithmic issues in game theory:

  • Computing equilibria / approximate equilibria in different kinds of games
  • Understanding quality of equilibria in load- balancing, networkā€“design, routing, machine schedulingā€¦
  • Analyzing dynamics of simple behaviors or adaptive (learning) algorithms: quality guarantees, convergence,ā€¦
  • Design issues: constructing rules so that game will (ideally) have dominant-strategy equilibria with good properties.

End of Game Theory Intro