Introduction to the Basics of Game Theory YuWen

Introduction to the Basics of Game Theory Yu-Wen Chen 2017 -12

Outline • Normal form game • Prison dilemma • Bayesian game • Auction • First-price sealed-bid auction • Second-price sealed-bid auction • Vickrey Clarke Groves mechanism

Normal Form Game • Prison dilemma: Player B C D Player A C D -1, -1 -3, 0 0, -3 -2, -2 • Each player has two pure strategies C: cooperate(with other player) D: defect • Dominant strategy: The strategy that produces the highest payoff of any strategy available for every possible action by the other players. Ex. The Dominant strategy for both player A and B is defect

Nash Equilibrium of Normal Form Game • Nash equilibrium: given all the strategies played by the other players , there is no strategy that a player could play that would yield a higher payoff. Player A C D Player C -1, -1 -3, 0 B D 0, -3 -2, -2 Prison dilemma • Mutual cooperation seems the best choice; however, each player could improve their own situation by betraying other player.

Bayesian Game • Bayesian game: a game in which the players have incomplete information on the other players • P: the probability that suspect is criminal P 1 -P Suspect is civilian Suspect’ s action Sheriff’s action Shoot Not Shoot -3, -1 -1, -2 Not -2, -1 0, 0 Suspect is criminal Suspect’ s action A Sheriff’s Dilemma Sheriff’s action Shoot Not Shoot 0, 0 2, -2 Not -2, -1 -1, 1

A Sheriff’s Dilemma • 1 -P Suspect is civilian Suspect’s Shoot action Not Sheriff’s action Shoot Not -3, -1 -1, -2 -2, -1 0, 0 P Suspect is criminal Suspect’s action Sheriff’s action Shoot Not Shoot 0, 0 2, -2 Not -2, -1 -1, 1

Bayesian (Nash) Equilibrium • Players choose strategies to maximize their payoffs in response to others accounting for 1. Strategic uncertainty about how others will play 2. Payoff uncertainty about the value to their actions • Explicitly models behavior in an uncertain environment

Auction • Auctions: any mechanisms for allocating resources among selfinterested agents • • • English auction Dutch auction First-price sealed-bid auction Second-price sealed-bid auction For each agent: Utility = valuation of the object - payment

First-price Sealed-bid Auction • Rules: • All bidders simultaneously submit sealed bids. • Agent with highest bid win • Winner pays Winner's bid • There is no dominate strategies • Bayesian-Nash equilibrium: • Suppose there are two bidders whose valuations a and b are drawn from a Continuous uniform distribution over the interval [0, 1] • When there are 2 agents, the Bayesian-Nash equilibrium is each bidder biding exactly half his/her value

First-price Sealed-bid Auction Analysis •

Second Price Sealed-bid Auction • Rules: • All bidders simultaneously submit sealed bids. • Agent with highest bid win • Winner pays second highest bid • Truth-telling is a dominant strategy

Second Price Sealed-bid Auction Analysis • Case 2 Case 1 Second-highest bid

Second Price Sealed-bid Auction Analysis • Case 4 Case 3 Winner pays Second-highest bid

DSIC Mechanism • Dominant-Strategy Incentive-Compatible • In a DSIC mechanism, strategic considerations cannot help any agent achieve better outcomes than the truthtelling • Therefore DSIC mechanism is also called strategy-proof or truthful.

Vickrey Clarke Groves (VCG) Mechanism •

VCG Example 1 Agent U (build road) U(do not build road) Payment 1 200 0 150 2 100 0 50 3 0 250 0 • Choose build road: 200 +100 + 0 > 0 + 250 • Payment of agent 1: If agent 1 is not exist: U(build road) =100 < U(do not build road) = 250 – 100 = 150 • Payment of agent 2: If agent 2 is not exist: U(build road) = 200 < U(do not build road) = 250 – 200 = 50 ✥Agent who make things worse for others by existing pays more than 0

VCG Example 1 (Cont. ) Agent U (build road) U(do not build road) Payment 1 200 0 150 2 100 0 50 3 0 250 0 • Choose build road: 200 +100 + 0 > 0 + 250 • Payment of agent 3: If agent 3 is not exist: U(build road) = 300 > U(do not build road) = 0 300 – 300 = 0 ✥Agent who don’t change the outcome pays 0

VCG Example 2 •

Proof of Truthfulness of VCG (1) •

Proof of Truthfulness of VCG (2) •

Proof of Truthfulness of VCG (3) •

VCG Limitation - Susceptibility to Collusion (1) Agents 1 and 2 both increase their declared by $50 • Payment of agent 1: 250 – 150 = 100 • Payment of agent 2: 250 – 250 = 0 Agent U (build) U(don’t build) Payment 1 200 0 150 1 250 0 100 2 100 0 50 2 150 0 0 3 0 250 0 Without collusion With collusion

VCG Limitation - Susceptibility to Collusion (2) • Payment of agent 2 without agent 3 exist: If agent 2 is not exist choose not to build road Payment of agent 2 = 90 – 0 = 90 • Payment of agent 2 with agent 3 exist: If agent 2 is not exist still choose to build road VCG make the same choice, therefore the payment of agent 2 is 0 Agent U (build) U(don’t build) Payment 1 0 90 0 2 100 0 90 2 100 0 0 - - 3 100 0 0 Without agent 3 With agent 3

References • • Wikipedia Game theory - Coursera Game theory II: Advanced Application - Coursera http: //auctiontheorycourse. wordpress. com/slides-and-notes/