efficiency and the redistribution of welfare milan vojnovic microsoft research cambridge, uk joint...
TRANSCRIPT
Efficiency and the Redistribution of
WelfareMilan Vojnovic
Microsoft ResearchCambridge, UK
Joint work with Vasilis Syrgkanis and Yoram Bachrach
2
Contribution Incentives
• Rewards for contributions• Credits• Social gratitude• Monetary incentives
• Online services• Ex. Quora, Stackoverflow, Yahoo! Answers
• Other• Scientific authorship• Projects in firms
3
Que
stion
Topi
c
Site
4
Another Example: Scientific Co-Authorship
5
Some Observations
• User contributions create value• Ex. quality of the content, popularity of the generated content
• Value is redistributed across users• Ex. Credits, attention, monetary payments
• Implicit and explicit signalling of individual contributions• Ex. User profile page, rating scores, etc• Ex. Wikipedia – not in an article, but by side means [Forte and Bruckman]• Ex. Author order on a scientific publication
6
How efficient are simple local value sharing schemes with respect to social welfare of the society as a whole?
7
Outline
• Game Theoretic Framework
• Efficiency of Monotone Games under a Vickery Condition
• Efficiency of Equal and Proportional Sharing
• Production Costs
• Conclusion
8
Utility Sharing Game
USG():• : set of players
• : strategy space,
• : utility of a player,
9
Project Contribution Games
Special: total value functions
1
2
i
n
1
2
j
m
Share of value
10
Monotone Games
• A game is said to be monotone if for every player and every
• It is strongly monotone, if for every player and every :
, for every
11
Importance of Monotonicity• ,
• Nash equilibrium condition
• Efficiency =
0 1
1
𝑥
𝑣 (𝑥)concave,
𝑏∗
12
Vickery Condition
• A game satisfies Vickery condition if for every player and :
• It satisfies k-approximate Vickery condition if for every player and :
]
Rewarded at least one’s marginal contribution
13
Local Value Sharing
• A project value sharing is said to be local if the value of the project is shared according to a function of the investments to this project:
, for every and
• Equal value sharing:
• Proportional value sharing:
14
DBLP database• 2,132,763 papers• 1,231,667 distinct authors • 7,147,474 authors
Scientific Co-Authorships
15
Scientific Co-Authorship (cont’d)
o random
16
Solution Concepts & Efficiency• Nash Equilibrium (NE)• Unilateral deviations
• Strong Nash Equilibrium (SNE)• All possible coalitional deviations
• Bayes Nash Equilibrium (BNE)• Incomplete information game
• Efficiency• Worst case ratio of social welfare in an equilibrium and optimal social welfare
17
Outline
• Game Theoretic Framework
• Efficiency of Monotone Games under a Vickery Condition
• Efficiency of Equal and Proportional Sharing
• Production Costs
• Conclusion
18
Efficiency in Strong Nash EquilibriumTheorem
• Any SNE of a monotone game that satisfies the Vickery condition achieves at least ½ of the optimal social welfare
• If the game satisfies the -approximate Vickery condition, then any SNE achieves at least of the optimal social welfare
19
Efficiency in Nash Equilibrium
Theorem
• Suppose that the following conditions hold:1) -approximate Vickery condition2) Strategy space of each player is a subset of some vector space3) Social welfare satisfies the diminishing marginal property
• Then, any NE achieves at least 1/() of the optimal social welfare
20
Local Vickery Condition
• A value sharing of a project is said to satisfy local k-approximate Vickery condition if
• If value sharing of all projects is locally k-approximate Vickery, then the value sharing is k-approximate Vickery
• Local k-approximate Vickery condition
}
𝜕𝑖𝑣 𝑗 (𝒃𝑗)
degree of substitutability
21
Degree of Substitutability
• If value functions satisfy diminishing returns property, then
• If , then each player is quintessential to producing any value, i.e. , for every
22
Degree of Substitutability (cont’d)
• Efficiency =
• If , then any local value sharing cannot guarantee a social welfare that is 1/ of the optimum social welfare
1
2
𝑛
1
2
𝑛−1
⋮𝑣1 (𝒃)=𝑛∏
𝑖
𝐼 (𝑏𝑖 ,1=1)
𝑣2 (𝒃 )=(1+𝜖) 𝐼 (𝑏𝑛 , 2=1)
Budget 1
}
23
Outline
• Game Theoretic Framework
• Efficiency of Monotone Games under a Vickery Condition
• Efficiency of Equal and Proportional Sharing
• Production Costs
• Conclusion
24
Equal Sharing
• Suppose that project value functions are monotone, then equal sharing satisfies the -approximate Vickery condition
25
Proportional Sharing
• Suppose that project value functions are functions of the total effort, increasing, concave, and
Then, proportional value sharing satisfies the Vickery condition
26
Proof Sketch
• concave and , for every
• Take and to obtain
27
Local Smoothness• A utility maximization game is -smooth iff for every and :
• A utility maximization game is locally -smooth iff with respect with respect to at which are continuously differentiable:
where
28
Efficiency of Smooth Games
• If a utility sharing game is locally ()-smooth with respect to a strategy profile then utility functions are continuously differentiable at every Nash equilibrium , then
29
Sufficient Condition for Smoothness• are concave functions of total effort, , and are continuously
differentiable • Proportional sharing of value • For all strategy profiles and and ,
Then, the game is locally -smooth with respect to
if , else
30
Efficiency by Smoothness:Fractional Exponent Functions
• Suppose that , and
• Then, proportional sharing achieves at least of the optimal social welfare
31
Efficiency by Smoothness:Exponential Value Functions
• Suppose , and
• Then, proportional value sharing achieves at least of the optimal social welfare
32
Tight Example
• is a Nash equilibrium where each player focuses all effort his effort on project 1
1
2
𝑖
𝑛
1
2
𝑛
𝑛−1}𝑣1 (𝑥 )=1−𝑒−𝛼𝑥
𝑣2 (𝑥 )=𝑞 (1−𝑒−𝛽𝑥 )
33
Tight Example (cont’d)• Nash equilibrium:
• Social optimum:
(players invest in distinct projects)
𝑢(𝒃)𝑢(𝒃∗)
→𝛼→∞,𝛽→0𝑛2
2𝑛2−2𝑛+1, large
1
2
𝑖
𝑛
1
2
𝑛
𝑣1
𝑣2}𝑛−1
34
Efficiency and Incomplete Information• Proportional sharing with respect to the observed contribution
• Concave value functions of the total contribution
• Abilities are private information
Then the game is universally -smooth, hence, in a Bayes Nash equilibrium, the expected social welfare is at least ½ of the expected optimum social welfare
35
Universal Smoothness
• Game • Value function • Game is -smooth with respect to the function if
for all types and and every outcome that is feasible under
[Roughgarden 2012, Syrgkanis 2012]
36
Efficiency under Universal Smoothness• Efficiency
• If a game is -smooth with respect to an optimal choice function then the expected social welfare is at least of the optimal social welfare
37
Outline
• Game Theoretic Framework
• Efficiency of Monotone Games under a Vickery Condition
• Efficiency of Equal and Proportional Sharing
• Production Costs
• Conclusion
38
Production Costs• Payoff for a player: • Social welfare , total value
production cost
∞
00
𝑐 𝑖 (𝑥 )
𝑥 Budget
constraint(earlier slides)
00
𝑐 𝑖 (𝑥 )
𝑥 Constant
marginal cost
00
𝑐 𝑖 (𝑥 )
𝑥 A convex increasing function
Examples
39
Elasticity
• Def. the elasticity of a function at is defined by
40
Efficiency
• Suppose that production cost functions are of elasticity at least and the value functions are of elasticity at most
• If is any pure Nash equilibrium and is socially optimal, then
Moreover
41
Efficiency (cont’d)
• Constant marginal cost of production is a worst case• But this is a special case: for any production cost functions with a
strictly positive elasticity, the efficiency is a constant independent of the number of players • Budget constraints are a best case
42
Conclusion• When the wealth is redistributed so that each contributor gets at least his
marginal contribution locally at each project, the efficiency is at least ½
• The degree of complementarity of player’s contributions plays a key role: the more complementary the worse
• Simple local value sharing• Equal sharing: the efficiency is at least 1/k, where k is the maximum number of
participants in a project• Proportional sharing: guarantees the efficiency of at least ½ for any concave project
value functions of the total contribution
• Production costs play a major function: the case of linear production costs is a special case for which the inefficiency can be arbitrarily small; at least a positive constant for any convex cost function of strictly positive elasticity