Lecture slides: Introduction to Game Theory, Slides of Game Theory

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Introduc)on to Game Theory

(JTGSS Lectures 1 and 2)

Bruce Hajek

Strategic form (i.e. normal form) games

• We’ll consider five examples
  • Second price auc)on
  • Conges)on game
  • Five ra)onal pirates and 100 gold coins
  • Rock-­‐scissors-­‐paper (and fic))ous play)
  • Prisoners’ dilemma (and repeated games) Players simultaneously each take an ac)on.

Some terminology

• A best response (BR) of a player to ac)ons by the

other player is one that maximizes her payoff

• A dominant strategy is a response that is a best

response for any ac)on of the other player

(may not exist)

• A vector of strategies (one for each player) is

efficient if it maximizes the sum of payoffs.

• A pair of strategies, such that each is a best

response for the other, is a Nash equilibrium.

• Claim: Bidding truthfully is a dominant strategy
  • Consider some bidder Alice
  • Let z=highest bid of the other bidders
  • Let x in V be the value of the object to Alice
  • If bidder Alice wins, she pays z; her payoff is x-­‐z
  • Alice’s profit is maximized if:
    • she wins if x>z, and
    • she loses if x<z
  • Happens if she bids x

Second price auc)on -­‐ con)nued

The dominant strategy equilibrium for second price auc)on is efficient. Doesn’t depend on Alice’s bid

7 A conges)on game (con)nued) Suppose ini)al routes are assigned arbitrarily. Then players, one at a )me, consider changing their route to a best response. payo↵ of player i: ⇡i(ri, ri) =

X

l 2 ri dl(nl) Iterated best response leads to monotone decreasing values of the poten)al func)on, with a strict decrease in poten)al func)on whenever a player changes route. Converges to a Nash equilibrium. Few games, however, admit a poten)al. Define the potential function (r 1 ,... , rn) =

X

l 2 L Dl(nl) where Dl(nl) =

P

nl k= dl(k). Then ⇡i(eri, ri) ⇡i(ri, ri) = (eri, ri) (ri, ri)

• Pirates A,B,C,D,E, ordered with most senior first, have to

decide how to share 100 gold coins

• The most senior pirate proposes how to share the coins
• All pirates vote on the proposal.
  • If at least half of the pirates vote yes, the proposal is implemented, game ends.
  • Otherwise, the first pirate is killed, and the remaining pirates engage in the same process.
• Each pirate would like to live. Given that, each pirate

would like to maximize number of coins he receives. If two

possibili)es have equal payoff, he’d prefer the one that kills

more pirates.

Example 3: Five Ra)onal Pirates and 100 Gold coins

• The dynamic programming solu)on yields the unique

subgame perfect Nash equilibrium

Five Ra)onal Pirates and 100 Gold coins (con)nued)

• Are there any other Nash equilibria?
• Yes. For example, suppose only C,D, and E remain, and E

insists that he will vote no unless C offers him 90 gold

coins. Suppose C does offer (-­‐,-­‐,10,0,90). C and E vote

to accept it. It is an NE for three pirates.

• The ul)matum by E, demanding 90 gold coins, is not a

credible threat, because if C does not offer him 90

coins, rejec)on by E would be irra)onal.

Strategic form (i.e. normal form) games

• Two players
• Each chooses R, P, or S
• P beats R, R beats S, S beats P Example 4: Rock -­‐ Paper – Scissors

Example: Paper-­‐Rock-­‐Scissors (con)nued) R P 0, 0 -­‐1, 1 1, -­‐ Player one Player two S P S R 0, 0 0, 0 1, -­‐ 1, -­‐ -­‐1, 1 -­‐1, 1 No. Keeps cycling. Moreover, there is no NE in pure strategies. Does iterated best response converge? Mixed strategies are probability distribu)ons over the ac)ons. Players seek to maximize expected gain.

Example: Paper-­‐Rock-­‐Scissors (con)nued) R P 0, 0 -­‐1, 1 1, -­‐ Player one Player two S P S R 0, 0 0, 0 1, -­‐ 1, -­‐ -­‐1, 1 -­‐1, 1 Mixed strategies are probability distribu)ons over the ac)ons. Players seek to maximize expected gain.

Unique NE is given by p 1 = p 2 =

1 3

1 3

1 3

⇡ 1 (p 1 , p 2 ) = p

T 1

M 1 p 2

⇡ 2 (p 2 , p 1 ) = p

T 2

M 2 p 1

where

M 1 =

A

; M 2 = M

T 1

16 (x) = 1 Z (e x 1 , e x 2 ,... , e xn ) i(pi) = (Mipi/⌧ ) i(pi) is BR for cost functions with entropy terms: U 1 (p 1 , p 2 ) = p T 1 M^1 p^2 +^ ⌧^ H(p^1 ) U 2 (p 2 , p 1 ) = p T 2 M^2 p^1 +^ ⌧^ H(p^2 ) The mollified best response dynamics becomes q ˙ 1 = 1 (q 2 ) q 1 q ˙ 2 = 2 (q 1 ) q 2 Let V 1 (q 1 , q 2 ) = maxs U 1 (s, q 2 ) U (q 1 , q 2 ) and V 2 (q 1 , q 2 ) = maxs U 1 (q 1 , s) U (q 1 , q 2 ). Find dV 1 (q 1 , q 2 ) dt  V 1 (q 1 , q 2 ) + ˙q T 1 M^1 q˙^2 dV 2 (q 1 , q 2 ) dt  V 2 (q 1 , q 2 ) + ˙q T 2 M 2 q˙ 1 So V 1 (q 1 , q 2 ) + V 2 (q 1 , q 2 ) is a Lyapunov function for proving global convergence.

Example: Rock-­‐Paper-­‐Scissors (con)nued) R P 0, 0 0, 1 1, 0 Player one Player two S P S R 0, 0 0, 0 1, 0 1, 0, 1 0, 1 A varia)on of Rock-­‐Paper-­‐Scissors is the nonzero some version. Fic))ous play does not converge (Shapley): R S R P R P S P S P S P S P S R S R S R S R S R S R

Example: Prisoners’ dilemma (con)nued) L L R R 2, -­‐1 0, 0 1, 1 -­‐-­‐1,

Prisoner

one

Prisoner

two

Some proper)es of prisoners’ dilemma

• R is a dominant strategy for each prisoner; (R,R) is a

dominant strategy equilibrium.

• The sum of payoffs for the dominant strategy pair is

• (L,L) (sum of payoffs = 1+1= 2) is the unique efficient

pair