Efficiency and the Redistribution of Welfare

Efficiency and the Redistribution of Welfare Milan Vojnovic Microsoft Research Cambridge, UK Joint work with Vasilis Syrgkanis and Yoram Bachrach

Contribution Incentives • Rewards for contributions • Credits • Social gratitude • Monetary incentives • Online services • Ex. Quora, Stackoverflow, Yahoo! Answers • Other • Scientific authorship • Projects in firms

Question Topic Site

Another Example: Scientific Co-Authorship

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

How efficient are simple local value sharing schemes with respect to social welfare of the society as a whole?

Outline • Game Theoretic Framework • Efficiency of Monotone Games under a Vickery Condition • Efficiency of Equal and Proportional Sharing • Production Costs • Conclusion

Utility Sharing Game USG(): • : set of players • : strategy space, • : utility of a player,

Project Contribution Games 1 1 Special: total value functions 2 2 i Share of value j n m

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

Importance of Monotonicity • , • Nash equilibrium condition • Efficiency = concave, 1 1 0

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

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:

Scientific Co-Authorships DBLP database • 2,132,763 papers • 1,231,667 distinct authors • 7,147,474 authors

Scientific Co-Authorship (cont’d) o random

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

Efficiency in Strong Nash Equilibrium Theorem • 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

Efficiency in Nash Equilibrium Theorem • Suppose that the following conditions hold: 1) -approximate Vickery condition 2) Strategy space of each player is a subset of some vector space 3) Social welfare satisfies the diminishing marginal property • Then, any NE achieves at least 1/() of the optimal social welfare

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 } degree of substitutability

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

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 1 2 } Budget 1

Equal Sharing • Suppose that project value functions are monotone, then equal sharing satisfies the -approximate Vickery condition

Proportional Sharing • Suppose that project value functions are functions of the total effort, increasing, concave, and Then, proportional value sharing satisfies the Vickery condition

Proof Sketch • concave and , for every • Take and to obtain

Local Smoothness • A utility maximization game is -smoothiff for every and : • A utility maximization game is locally-smoothiff with respect with respect to at which are continuously differentiable: where

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

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

Efficiency by Smoothness:Fractional Exponent Functions • Suppose that , and • Then, proportional sharing achieves at least of the optimal social welfare

Efficiency by Smoothness:Exponential Value Functions • Suppose , and • Then, proportional value sharing achieves at least of the optimal social welfare

} Tight Example 1 1 2 2 • is a Nash equilibrium where each player focuses all effort his effort on project 1

Tight Example (cont’d) } 1 1 2 2 • Nash equilibrium: • Social optimum:(players invest in distinct projects) , large

Efficiency and Incomplete Information • Proportional sharing with respect to the observed contribution • Concave value functions of the total contribution • Abilities are private informationThen 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

Universal Smoothness • Game • Value function • Game is -smooth with respect to the function iffor all types and and every outcome that is feasible under [Roughgarden2012, Syrgkanis 2012]

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

Production Costs production cost • Payoff for a player: • Social welfare , total value Examples Constant marginal cost A convex increasing function Budget constraint (earlier slides)

Elasticity • Def. the elasticity of a function at is defined by

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

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

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

Efficiency and the Redistribution of Welfare