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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∗
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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
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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
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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
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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
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Proof Sketch
• concave and , for every
• Take and to obtain
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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
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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
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Efficiency by Smoothness:Fractional Exponent Functions
• Suppose that , and
• Then, proportional sharing achieves at least of the optimal social welfare
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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