Yair Zick Joint work with Yoram Bachrach, Ian Kash and Peter Key

 The school of computer science must decide on its 2015 budget allocation. How should the budget be divided? Dean’s Business Office Office of Alumni Relations SCS Career Center

Dean’s Business Office Computer Science Dpt. $50M $40M $60M $50M $70M $55M Helpdesk$10M $15M  Budget division on one year will affect revenue on the next.  What is the best way to divide revenue?

 Key Observation: the way profits are divided affects revenue in subsequent rounds.  An underfunded department will not be as efficient.  Extra funding may not lead to extra benefit.  Limitation: individual departments do not care about total utility, want to maximize their own share of the budget.

 Find a sequence of budget divisions (a contract) that maximizes total utility up to time-step T (optimal contract at time T)  Given an optimal contract at time T, how different is it from a contract that maximizes individual utility at time T?

 Agents who care about their future revenue tend to be more collaborative (will prefer revenue divisions that are near optimal)  Agents who are invested in others are more collaborative.  Homogeneous utility functions.

t =0123 w1w1 w 1 (0) x1V1x1V1 x1V2x1V2 x1V3x1V3 w2w2 w 2 (0) x2V1x2V1 x2V2x2V2 x2V3x2V3 total - V 1 = v ( w (0)) V 2 = v ( x 1 V 1, x 2 V 1 ) V 3 = v ( x 1 V 2, x 2 V 2 ) Players want to maximize their utility Social Welfare of the contract  Utility Function - v : R 2 ! R +  Initial Endowments – w (0) = ( w 1 (0), w 2 (0))  Sharing Contract - ( x 1, x 2 ) s.t. x 1 + x 2 = 1

v ( x, y ) = 4 xy w (0) = (0.5,0.5) t =01234 w1w1 0.5 w2w2 (0.5,0.5): t =01234 w1w1 0.5¾(¾) 2 (¾) 8 (¾) 16 w2w2 0.5¼¼∙¾¼∙(¾) 7 ¼∙(¾) 15 (3/4,1/4): … If player 1 only cares about utility up to time 2, then the second contract is better. Otherwise....

 First objective: find a sequence of revenue divisions that maximize revenue up to time T  Theorem: if the utility function is homogeneous, there exist optimal stationary contracts: contracts that offer the same revenue division at all time steps.  In fact: these contracts are precisely the maxima of v ( x )

 A function is homogeneous of degree k if v ( cx ) = c k v ( x )  If v is homogeneous, then it’s much easier to work with:  finding an optimal contract  finding total revenue at time t (easily derive a closed-form rather than recursive formula).

 Are there stationary contracts that are both optimal and individually optimal? (i.e. no agent wants to change them)  If v is differentiable, NO – an inevitable tension between individual gain and social welfare.  But…

individually optimal contracts become “nearly” optimal

individually optimal contracts are not optimal…

 First case: allow individuals to choose any revenue division they want at each round.  Fix a time step q ; let x q ( T ) be the best contract for player i at time q, if his goal is to maximize total revenue up to time T

 Theorem: let x q ( T ) be the best contract for player i at time q, if his goal is to maximize total revenue up to time T. If the utility function is  Differentiable  Homogeneous of degree 1 or more then lim T !1 x q ( T ) exists and is a critical point of v If v is concave, then the limit is an optimal contract.

 Theorem: If the utility function is  Differentiable  Homogeneous of degree 1 or more then lim T !1 x q ( T ) exists and is a critical point of v If v is concave, then the limit is an optimal contract.

 Theorem: if we limit individuals to picking fixed contracts (i.e. the same revenue division at all time-steps), then as their horizon increases, the contracts they will pick converge to an optimal contract.  Conclusion: far-sighted agents understand that what’s best for the group is also (nearly) best for them!

 The setting we describe is not a game (no player actions)  But – we can easily derive strategic games  utility function depends on players’ private information.  Each player proposes a contract, center aggregates contracts. Our results show that non-myopic agents will be much less strategic!

 When considering long-term gains, fair payoff divisions can be reached.  Optimal contracts (for differentiable production functions) are never individually optimal  … but can be so in the limit.

 Possible applications:  Networks, weighted matching, exchange markets…  Agents with divisible resources  Uncertain environments?  Beyond resource allocation settings?

 Revenue Reinvestment: we assume that each agent invests all of his revenue back into the function  reasonable, if we assume agents are (non-corrupt) institutes.  We can allow agents to keep a constant share of the profits, everything carries through.  If we allow strategic reinvestments (you choose how much to put back for the next round), things get interesting.  Non-myopic agents will invest everything into the function in early rounds, reap rewards at later rounds.