NON-MYOPIC COLLABORATORS (NEARLY) GET THEIR WAYYair Zick

Joint work with Yoram Bachrach, Ian Kash and Peter Key

MOTIVATIONThe 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

MOTIVATION

2015 2016 2017

Computer Science Dpt.

$50M

$40M

$60M

$50M

$70M

$55M

Helpdesk $10M $10M $15M

Budget division on one year will affect revenue on the next.

What is the best way to divide revenue?

MOTIVATIONKey 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.

OBJECTIVEFind 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?

OUR RESULTSAgents 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.

OUR SETTING

t = 0 1 2 3

w1 w1(0) x1V1 x1V2 x1V3

w2 w2(0) x2V1 x2V2 x2V3

total - V1 = v(w(0)) V2 = v(x1V1,x2V1) V3 = v(x1V2,x2V2)

Players want to maximize their utility

Social Welfare of the contract

Utility Function - v: R 2! R+

Initial Endowments – w(0) = (w1(0), w2(0))

Sharing Contract - (x1, x2) s.t. x1 + x2 = 1

AN EXAMPLEv(x,y) = 4xy w(0) = (0.5,0.5)

t = 0 1 2 3 4

w1 0.5 0.5 0.5 0.5 0.5

w2 0.5 0.5 0.5 0.5 0.5(0.5,0.5):

t = 0 1 2 3 4

w1 0.5 ¾ (¾)2 (¾)8 (¾)16

w2 0.5 ¼ ¼∙¾ ¼∙(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....

OPTIMAL CONTRACTSFirst 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)

OPTIMAL CONTRACTSA function is homogeneous of degree k if

v(cx) = ckv(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).

OPTIMAL VS. INDIVIDUALLY OPTIMALAre 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…

OPTIMAL VS. INDIVIDUALLY OPTIMAL (both players start with ½)

individually optimal

contracts become “nearly”

optimal

OPTIMAL VS. INDIVIDUALLY OPTIMAL (both players start with 1)

individually optimal

contracts are not optimal…

CONVERGENCE RESULTSFirst case: allow individuals to choose any revenue division they want at each round.

Fix a time step q; let xq(T) be the best contract for player i at time q, if his goal is to maximize total revenue up to time T

CONVERGENCE RESULTSTheorem: let xq(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 DifferentiableHomogeneous of degree 1 or morethen limT !1 xq(T) exists and is a critical point of vIf v is concave, then the limit is an optimal contract.

CONVERGENCE RESULTSTheorem:If the utility function is DifferentiableHomogeneous of degree 1 or morethen limT !1 xq(T) exists and is a critical point of vIf v is concave, then the limit is an optimal contract.

CONVERGENCE RESULTSTheorem: 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!

A NOTE ON STRATEGIC BEHAVIOR

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!

CONCLUSIONS AND FUTURE WORKWhen 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.

CONCLUSIONS AND FUTURE WORKPossible applications:

Networks, weighted matching, exchange markets…

Agents with divisible resourcesUncertain environments? Beyond resource allocation settings?

THANK YOU! QUESTIONS?

EASTER EGGSIndividual Utility Functions: instead of having one utility function , we have that the utility of agent is given by , the center wants to choose contracts that maximize , but individuals want contracts that maximize their own utility.

EASTER EGGSDiscount factors: for a large enough discount factor, our results carry through (revenue grows really fast if the functions are homogeneous of degree )

Individual utility functions

EASTER EGGS 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.