Communication Networks A Second Course Jean Walrand Department of EECS University of California at Berkeley

Games in Networks: Overview Motivation Examples of Game In Networks Key Concepts

Motivation Decisions made by people (agents) Selfish agents with conflicting interests Markets (interactions of agents) influence technological development and adoption Protocols limit or enable strategies Design protocols to create suitable incentives Pricing, service choice, contracts, regulations, cooperation, mergers and acquisitions, liability Game theory models interactions of selfish agents

Examples of Games Matching Pennies: Alice and Bob both show a penny If the faces match, Bob gives Alice $1.00 If the face do not match, Alice gives Bob $1.00 Battle of the Sexes: Alice and Bob choose to go to the opera or football They prefer to go together, but Alice loves opera and hates football; Bob loves football and hates opera Prisoners’ Dilemma: Alice and Bob are accused of a crime and are interrogated separately If exactly one pleads guilty, that person gets 1 year and the other five years If both plead guilty, they get 3 years; if none pleads guilty, they get 2 years.

In Networks Physical MAC Routing Transport Interactions between providers

Games: Physical Layer Being Loud in a Restaurant You are heard better if you talk louder Everybody ends up being very loud and upset by the noise Power control in CDMA Each user prefers to transmit at maximum power to improve his capacity However a user’s increased power reduces the capacity of the other users because of increased interference Spectrum allocation Each user prefers a wider spectrum However, this decreases the capacity of the other users

Games: MAC Backup Window in WiFi Each user prefers a smaller backup window size However, this reduces the throughput of others Persistence in Aloha Similar Priority Class in 11e Incentive is to claim high priority for all traffic However, this reduces the quality …

Games: Routing Shortest Path Users may prefer shortest path However, this my lead to congested paths Fastest Paths Users may prefer fastest path This may increase average delay Duplicated Paths Users may prefer to duplicate their traffic across parallel paths This may make everybody worse off Hot potato routing Send packets off to competitor’s network asap

Games: Transport TCP Users might want to be more aggressive If everyone does it, everybody is worse off Proportional Fairness Normal: user gets x = w/p Strategic: user knows p = w + other weights This leads strategic users to change their bids….

Games: Between Providers Interactions between network providers User ---[AT&T]----[Verizon]--- User What are the incentives for Verizon to provide good service to AT&T’s clients? How should one allocate profits? User ---[AT&T]---[Google] What are the incentives for AT&T to provide good service to Google? Should AT&T be allowed to charge Google more for some services?

Key Concepts Game: Multiple Agents Agents act in some order, possibly multiple times When she acts, agent knows some information about previous actions, possibly incomplete Ultimate reward of each agent depends on the actions of all the agents Agents may have incomplete information about each other and the game evolution may be random The choice of the actions may be randomized

Key Concepts Game Example: Matching Pennies HT H1, -1-1, 1 T 1, -1 Actions of Alice Actions of Bob Reward of Alice Reward of Bob One-Shot Game Both players play simultaneously Each player knows both reward functions One-Shot Game Both players play simultaneously Each player knows both reward functions

Key Concepts Game Example: Battle of the Sexes One-Shot Game Both players play simultaneously Each player knows both reward functions One-Shot Game Both players play simultaneously Each player knows both reward functions OF O4, 12, 2 F0, 01, 4 O = Opera F = Football

Key Concepts Game Example: Prisoners’ Dilemma One-Shot Game Both players play simultaneously Each player knows both reward functions One-Shot Game Both players play simultaneously Each player knows both reward functions G = plead guilty NG = plead not guilty GNG G3, 31, 5 NG5, 12, 2

Key Concepts Game Example: Dynamic 2-player, 3-step game: P1, P2, P1 P1 does not see the action of P2 Each player knows both reward functions 2-player, 3-step game: P1, P2, P1 P1 does not see the action of P2 Each player knows both reward functions

Key Concepts Game Example: Cournot Duopoly One-shot game Continuous action space Each player knows both reward functions One-shot game Continuous action space Each player knows both reward functions Two firms produce quantity q 1 and q 2 of a product The price is A – q 1 – q 2 For i = 1, 2 the profit of firm i is q i (A – q 1 – q 2 ) - Cq i Two firms produce quantity q 1 and q 2 of a product The price is A – q 1 – q 2 For i = 1, 2 the profit of firm i is q i (A – q 1 – q 2 ) - Cq i

Key Concepts Game Example: Bayesian One-shot game Continuous action space Each player has an incomplete knowledge of some reward functions One-shot game Continuous action space Each player has an incomplete knowledge of some reward functions Two firms produce quantity q 1 and q 2 of a product The price is A – q 1 – q 2 For i = 1, 2, the profit of firm i is q i (A – q 1 – q 2 ) – C i q i For i = 1, 2, firm i knows C i but only knows the distribution of C 2 - i Two firms produce quantity q 1 and q 2 of a product The price is A – q 1 – q 2 For i = 1, 2, the profit of firm i is q i (A – q 1 – q 2 ) – C i q i For i = 1, 2, firm i knows C i but only knows the distribution of C 2 - i

Key Concepts Strategy: Rule that specifies how to act at any time, as a function (possibly randomized) of the information available Good Strategy? Dominant Iterated deletion of dominated strategies Nash Equilibrium Other concepts … later.

Key Concepts: Good Strategy Dominant Strategy: Best, no matter what the other players do Example: Being loud in a restaurant. (Hum, it all depends on the reward function, of course.) Counter-example: The best response generally depends on the action of others. Think matching pennies – There is no dominant strategy

Key Concepts: Good Strategy Iterated deletion of dominated strategies: Example: Guess 2/3 of average Let us all choose a number in {0, 1, …, 100} The person who is closest to 2/3 of the average of all the numbers wins Let us all choose a number in {0, 1, …, 100} The person who is closest to 2/3 of the average of all the numbers wins Average is at most 100 Nobody should guess more than 66 2/3 average is then at most 44 Nobody should guess more than 44 2/3 average is then at most 30 Nobody should guess more than 30 2/3 average is then at most 20 … Only reasonable guess is 0 Average is at most 100 Nobody should guess more than 66 2/3 average is then at most 44 Nobody should guess more than 44 2/3 average is then at most 30 Nobody should guess more than 30 2/3 average is then at most 20 … Only reasonable guess is 0 Note: Assumes all players are very rational ….

Key Concepts: Good Strategy Iterated deletion of dominated strategies: Example:

Key Concepts: Good Strategy Nash Equilibrium: No one wants to deviate unilaterally Examples: GNG G3, 31, 5 NG5, 12, 2 Prisoners’ Dilemma HT H1, 00, 1 T 1, 0 Matching Pennies : 50/50 VW V1, 10, 0 W 1, 1 OF O4, 12, 2 F0, 01, 4 Coordination GameBattle of the Sexes

Key Concepts: Good Strategy Nash Equilibrium: No one wants to deviate unilaterally Examples: GNG G3, 31, 5 NG5, 12, 2 Prisoners’ Dilemma HT H1, 00, 1 T 1, 0 Matching Pennies : 50/50 VW V1, 10, 0 W 1, 1 OF O4, 12, 2 F0, 01, 4 Coordination GameBattle of the Sexes