Game Theory MOT Seminar John Musacchio johnm@soe.ucsc.edu 4/16/09 1
Who am I? John Musacchio Assistant Professor in ISTM Joined January 2005 PhD from Berkeley in Electrical l Engineering i Research Interests Game Theory applied to pricing problems. Modeling and control of networks Experience 2½ years at a Start-Up Helped design a chip-set for computer-networking switches 2
What is Game Theory? Study of interacting strategic agents. Used frequently in economics and other sciences. Competition between firms. Auction Design. International Policy. Evolution of Species. And many more 3
User Discrimination: Good or Bad? Content Provider A??? Content Provider B ISP 1 ISP 2 ISP 2 needs to invest To enable A s service
Net Neutrality: Issues??? A Should A have to pay ISP 2? B ISP 1 ISP 2 Content providers pay their ISP Would allowing 2 to charge A encourage 2 to invest? discourage A to invest? What revenue sharing mechanisms should new Internet have? John Musacchio
Classic Example: Prisoner s Dilemma Prisoner A Silent Betray Prisoner B Silent (-1,-1) 1) (-4,0) Betray (0,-4) (-3,-3) 3) 8
Competing Firms Firm A Firm B Stagnate Innovate Stagnate (2,2) 2) (-1,3) Innovate (3,-1) (0,0) 0) 9
Elements of a Game Players Strategy A Player s action Innovate or Stagnate Strategy Space Set of all possible actions Strategy Profile Particular combination of player strategies. Payoff A mapping from player strategy profile to player rewards Example: U( (I, I) ) = (0,0) 10
Solution Concept Nash Equilibrium A strategy profile from which no player has an incentive to deviate unilaterally Example (I,I) is a NE U A (I, I) > U A (S, I) Firm A cannot do better by deviating U B (I, I) > U B (I, S) Firm B cannot do better by deviating (I, I) is a Nash Equilibrium. Do all games have a Nash Equilibrium? 11
Example: Leader, Imitator (matching pennies) Idea: Player 1 (Imitator) wants a match, Player 2 (Leader) doesn t. What is the Nash Equilibrium? Expand strategy space to allow randomized or mixed strategies. Product A Leader Product B Imita ator Leader Deviates Product A (1,-1) (-1, 1) Imitator Deviates Leader Deviates Product B (-1,1) (1,-1) Imitator Deviates 12
Example: Leader, Imitator 1 Lead der Probabilit ty of A 0 Leader s Best Response 0 1 Imitator Probability of A NE is not strict in this case. Imitator s Best Response Nash Equilibrium (0.5, 0.5) (At NE, players are indifferent to switching) Such an NE is said to be not strict 13
Nash Existence Finite Strategy Space (J.F. Nash 1950) Every n-player game has at least one Nash Equilibrium (possibly mixed). 14
Static vs. Multi Stage Static Games Players choose strategies simultaneously, without knowing what the others do. Multi-Stage Game is played in multiple rounds. Players may see how others played in previous rounds. That information helps choose how to play in the next round. A strategy is a full specification of what actions to take in each stage, as a function of the observations from previous stages 15
Repeated Innovating Firms Game (Repeated Prisoners Dilemma) Firm A Firm B Recall, Both firms innovate in the one-shot game. Combined reward is 0. If they had both stagnated instead, their combined reward would have been 4. What happens if the game is repeated? Same game is repeated every year forever. NPV of future payoffs is discounted by a discount factor β. 16
Aside: What is Net Present Value? How much is 1 worth to you 1 year from now? Something less than a 1 say β. β 1 = Year 0 1 Year 0 1 β β 2 β n 3 = Year... n-2n-1 Year 0. n-3 Year 0 n 1 17
Repeated Innovating Firms Game (Repeated Prisoners Dilemma) Firm A s: Firm B s Where x n = Albert s action in slot n A( ) = Albert s payoff function y n = Bob s action in slot n B( ) = Bob s payoff function β = Discount factor 18
Repeated Innovating Firms Game Strategy: I will stagnate as long as you do. Threat: If you choose to innovate once, I will innovate forever thereafter. This is a NE if β > 1/3 19
Repeated Innovating Firms Game Proof: Suppose at time t, A innovates B retaliates by innovating forever thereafter A is forced to innovate at times t+1, t+2, as well A s As net payoff One Step Reward Future Consequences 20
Repeated Innovating Firms Game Intuition When β is large, future consequences of breaking the collusion agreement outweigh short term gain. When β is small, short term gain is more important than long term consequences. 21
SPE Player 2 L (2,2) 2) Player 1 L R L (1,10) R (-1,-1) 1) L R (2,2) (-1,-1) R (1,10) (1,10) Player 2 can threaten to choose R in stage 2 to get Player 1 to pick R is stage 1. But in the subgame starting in slot 2, Player 2 is compelled to pick L. Player 2 s threat is not credible. (R,R) is indeed a NE, but not SPE. Only (L,L) L) is a SPE. 22
Repeated Innovating Firms Game We said that the following strategy profile is a Nash Equilibrium: Strategy: I will stagnate as long as you do. Threat: If you choose to innovate once, I will innovate forever thereafter. Is it a SPE? Yes. In the subgame after the first deviation, it is rational to Innovate forever thereafter if you expect your opponent to do the same. 23
Repeated Innovating Firms Game Folk Theorem Cooperating can be rational if games are repeated forever. However, threat strategies can be used to enforce other outcomes. (3,-1) (2,2) Claim: Any Reward vector In the green region can be enforced by an SPE,. (0,0) (-1,3) 24
Repeated Innovating Firms Game Proof: Consider v in the green region: v (0,0) Pick integers N j that satisfy One shot rewards. They agree to play (I,I) I) the first N 1 steps (S,S) the next N 2 steps, etc When someone deviates from the schedule, the other retaliates by playing I forever thereafter. 25
A Two-Sided Market Analysis of Provider Investment Incentives With an Application to the Net-Neutrality N Issue. John Musacchio Technology and Information Management University of California, Santa Cruz johnm@soe.ucsc.edu http://www.soe.ucsc.edu/~johnm Galina Schwartz and Jean Walrand EECS University of California, Berkeley
Net-Neutrality Dimensions of Debate Offering of grades of service Freedom of speech Whether Local ISPs should be allowed to charge content providers
Overview Content Provider A??? Content Provider B ISP 1 ISP 2 ISP 2 needs to invest To enable A s service
Overview A??? Should A have to pay ISP 2? B ISP 1 ISP 2 Content providers pay their ISP Would allowing 2 to charge A encourage 2 to invest? discourage A to invest?
Neutral Network ISP Content Provider? ISP ISP Content provider connects to cheapest ISP(s) Any such connection allows communication with all end-users Competition drives connection prices to marginal cost We normalize so that Content providers pay 0 for connection
Non Neutral Network Content Provider ISP All ISPs can charge the content provider Content provider forced to pay all ISPs that serve end users.
Two-Sided (Non-Neutral) Neutral) One-Sided (Neutral) Advertisers C Advertisers 1 C U 1 1 U 1 T 1 T 1 C M T N U N C M T N U N Which is better? Study Investment Incentives Model Overview usage ( clicks ) function of provider investments Provider revenue function of usage and regime (one- vs. two-sided) d) Content and transit providers play a game
Two-Sided Markets Large Literature Idea See Rochet and Tirole (2006) for overview Platform mediating two types of participants E.g. Videogame Console needs to attract end-users and game makers Novelty of our model Model Investment incentives to compare two regimes. Previous application to Net-Neutrality Neutrality issue Hogedorn (2006) conduits, service providers, content Study open access of conduits by to service providers
Neutral Case Single Provider Advertisers C 1 T 1 U 1 Invest: t 1 Set user price: p 1 Invest: c 1 Click rate: Advertisers Advertisers pay C 1 B 1 a Users pay T 1 C U 1 1 T 1 B 1 p 1
Non-Neutral Case Single Provider Advertisers C 1 T 1 U 1 Invest: t 1 Set user price: p 1 Content provider charge: q 1 Invest: c 1 Click rate: Advertisers Advertisers pay C 1 C 1 pays T 1 C 1 B 1 a B 1 q 1 T 1 Users pay T 1 B 1 p 1 U 1
Neutral Case Multi Provider Advertisers C 1 U T 1 1 C M T N Invest: t n Set user price: p n Invest: c m Click rate on Tn: U N Advertisers Value of Content T s Investment Across Internet n Spillover of Other Transit Investment Advertisers pay C m Users pay T n : C m Tn U n
Non-Neutral Case Multi Provider Advertisers C 1 U T 1 1 C M T N Invest: t n Set user price: p n Set Content price: q n U N Invest: c m Click rate on T n : Advertisers pay C m C m pays T n Users pay T n : Advertisers C m T U U n T n
Net Payoff to Providers Payoff to each Transit provider [Revenue] - α [investment] Payoff to each Content Provider [Revenue] [Payment to Transit] β [Investment]
Game Theoretic Analysis Backwards Induction. Content Provider Strategy Evaluate first order condition of partial derivative w.r.t. c m Find symmetric equilibrium where c m s m=1 M are the same Transit Provider Strategy Evaluate first order condition of partial derivatives w.r.t. t n, p n, q n Find Symmetric equilibrium
Non-Neutral Case
Neutral Case
Social Welfare Sum of transit, content, and user welfare Welfare = payoff for providers For users, find consumer surplus: clicks price
Neutral vs Non Neutral for each Provider Type Transit Provider Payoff Content Provider Payoff a/θ a/θ
Comparison
Comparison N= Number Transit providers Neutral Better Non-Neutral N Better T pays C Non-Neutral Better C pays T [advertising rate] : [end user price sensitivity] J. Musacchio, G. Schwartz, J. Walrand, Network Neutrality and Provider Investment Incentives, in submission (2007).
Comparison
Price charged by Transit Provider to Content t Provider (Non Neutral)
Castles Toll: q 1 Toll: q 2 Toll: q 3 Toll: q 4 Tolls collected are a product of toll rate and traffic rate, A castle sees any benefit of his toll increase, but the downside (the traffic decrease) is borne by all castles. Consequently, each castle tends to tax higher than would be optimum socially.
Conclusions Two competing effects Need to adjust revenue sharing between content and transit providers. Castles on the Rhine effect of transit providers charging higher than optimal tolls. Whether neutral or non-neutral is better depends on number of providers advertising rates vs. user price sensitivity For parameters that make non-neutral superior, both content and transit providers are better off!
References Rochet, J.-C. and J. Tirole (2006) Two-Sided Markets: A Progress Report, RAND Journal of Economics, 37(3): 645-667. Hermalin, B. and M. Katz (2007), The Economics of Product-Line Restrictions With an Application to the Network Neutrality Debate, Competition Policy Center Hogendorn, C. (2007) Broadband Internet: net neutrality versus open access, International Economics and Economic Policy, 4: 185-208.