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# Mechanism design theory examples and complexity

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• 1. Mechanism Design Theory: How to Implement Social Goals Examples and Algorithmic Design Stathis Grigoropoulos Ioannis Katsikarelis Multi-Agent Systems Utrecht University 2012
• 2. Contents &#x2022; &#x2022; &#x2022; &#x2022; &#x2022; Introduction Context Examples Algorithmic Design Conclusion
• 3. Introduction Game Theory : A Normal Game Form[1] &#x2013; A Set of Players: Who is involved? &#x2013; A Set of Rules (Institutions): What can the players do? &#x2013; A Set of Outcomes: What will happen when players perform a certain action? &#x2013; A Set of Preferences: What players want the most of the possible outcomes? The importance of Game Theory is established in numerous fields: &#x2013; Economic Phenomena &#x2013; Auctions, Bargains, Fair Division &#x2013; Social Phenomena &#x2013; Social Network Formation, Social Choice &#x2013; Political Sciences &#x2013; Election outcome, Political choices
• 4. Introduction A closer look at a &#x201C;real&#x201D; case[2]: &#x2022; You are selling a rare painting for which you want to raise the maximum revenue. &#x2013; Tyler, who values the painting at \$100,000 &#x2013; Alex who values it at \$20,000 &#x2022; You do not know their valuation, how to get max revenue? &#x2013; Auction! &#x2013; If you knew, you would set the price at \$99,999 and be fine! &#x2022; Standard English open-cry Auction &#xF0E0; \$20,001 (not max!) &#x2022; + Reserve Price say at \$50,000 &#xF0E0; \$50,001 (close, but..lucky!) &#x2022; But why stop at a reserve price? How about a reserve price and an entry fee? But why stop at reserve prices and entry fees?....
• 5. Context The design of the institutions (Rules) through which individuals (Players) interact can have a profound impact on the results (Outcomes) of that interaction[3]. We saw: &#x2022; Auction is conducted with sealed bids versus oral ascending bids can have an impact on what bidders learn about each other's valuations and ultimately how they bid. &#x2022; Wanted: forecast the economic or social outcomes that these institutions generate
• 6. Context Theory of Mechanism Design[4] &#x201C;engineering&#x201D; part of economic theory &#x2022; much of economic theory devoted to: &#x2013; understanding existing economic institutions &#x2013; explaining/predicting outcomes that institutions generate &#x2013; positive, predictive &#x2022; mechanism design &#x2013; reverses the direction &#x2013; begins by identifying desired outcomes (goals) &#x2013; asks whether institutions (mechanisms) could be designed to achieve goals &#x2013; if so, what forms would institutions take? &#x2013; normative, prescriptive - - i.e., part of welfare economics
• 7. Examples Simple example[5], suppose &#x2022; mother wants to divide cake between 2 children, Alice and Bob &#x2022; goal: divide so that each child is happy with his/her portion &#x2013; Bob thinks he has got at least half &#x2013; Alice thinks she has got at least half call this fair division &#x2022; If mother knows that the kids see the cake in same way she does, simple solution: &#x2013; she divides equally (in her view) &#x2013; gives each kid a portion
• 8. Examples &#x2022; But what if, say, Bob sees cake differently from mother? &#x2013; she thinks she&#x2019;s divided it equally &#x2013; but he thinks piece he&#x2019;s received is smaller than Alice&#x2019;s &#x2022; difficulty: mother wants to achieve fair division &#x2013; but does not have enough information to do this on her own &#x2013; in effect, does not know which division is fair
• 9. Examples &#x2022; Can she design a mechanism (procedure) for which outcome will be a fair division? (even though she does not know what is fair herself ?) &#x2022; Age-old problem &#x2013; Lot and Abraham dividing grazing land
• 10. Examples Age-old solution: &#x2013; have Bob divide the cake in two &#x2013; have Alice choose one of the pieces Why does this work? &#x2022; Bob will divide so that pieces are equal in his eyes &#x2013; if one of the pieces were bigger, then Alice would take that one &#x2022; So whichever piece Alice takes, Bob will be happy with other &#x2022; And Alice will be happy with her own choice because if she thinks pieces unequal, can take bigger one
• 11. Examples Example illustrates key features of mechanism design: &#x2022; mechanism designer herself doesn&#x2019;t know in advance what outcomes are optimal &#x2022; so must proceed indirectly through a mechanism &#x2013; have participants themselves generate information needed to identify optimal outcome &#x2022; complication: participants don&#x2019;t care about mechanism designer&#x2019;s goals &#x2013; have their own objectives &#x2022; so mechanism must be incentive compatible &#x2013; must reconcile social and individual goals
• 12. Examples Example from the paper Consider society with &#x2022; 2 consumers of energy &#x2013; Alice and Bob &#x2022; Energy authority &#x2013; must choose public energy source &#xF0A7; &#xF0A7; &#xF0A7; &#xF0A7; gas oil nuclear power coal
• 13. Examples Two states of world State 1 consumers weight future lightly (future relatively unimportant) state 2 consumers weight future heavily (future relatively important) Alice &#x2013; cares mainly about convenience In state 1: favors gas over oil, oil over coal, and coal over nuclear In state 2: favors nuclear over gas, gas over coal, and coal over oil &#x2212; technical advances expected to make gas, coal, and especially nuclear easier to use in future compared with oil Bob &#x2013; cares more about safety In state 1: favors nuclear over oil, oil over coal, and coal over gas In state 2: favors oil over gas, gas over coal, and coal over nuclear &#x2212; disposal of nuclear waste will loom large &#x2212; gas will become safer
• 14. Examples State 1 Alice gas oil coal nuclear Bob nuclear oil coal gas State 2 Alice nuclear gas coal oil Bob oil gas coal nuclear &#x2212; energy authority &#xF0A7; wants source that makes good compromise between consumers&#x2019; views &#xF0A7; so, oil is social optimum in state 1 &#xF0A7; gas is social optimum in state 2 &#x2212; but suppose authority does not know state &#xF0A7; then does not know whether oil or gas better
• 15. Examples State 1 State 2 Alice Bob gas nuclear oil oil coal coal nuclear gas oil optimal Alice Bob nuclear oil gas gas coal coal oil nuclear gas optimal &#x2212; authority could ask Alice or Bob about state &#x2022; but Alice has incentive to say &#x201C;state 2&#x201D; regardless of truth always prefers gas to oil gas optimal in state 2 &#x2022; Bob always has incentive to say &#x201C;state 1&#x201D; always prefers oil to gas oil optimal state 1 So, simply asking consumers to reveal actual state too naive a mechanism
• 16. Examples State 1 State 2 Alice Bob gas nuclear oil oil coal coal nuclear gas social optimum: oil Alice Bob nuclear oil gas gas coal coal oil nuclear social optimum: gas Authority can have consumers participate in the mechanism given by table Bob Alice oil coal nuclear gas &#x2022; Alice &#x2013; can choose top row or bottom row &#x2022; Bob &#x2013; can choose left column or right column &#x2022; outcomes given by table entries &#x2022; If state 1 holds Alice will prefer top row if Bob plays left column Bob will always prefer left column so (Alice plays top, Bob plays left) is Nash equilibrium neither participant has incentive to change unilaterally to another strategy &#x2212; so good prediction of what Alice and Bob will do
• 17. Examples State 1 State 2 Alice Bob gas nuclear oil oil coal coal nuclear gas social optimum: oil Alice Bob nuclear oil gas gas coal coal oil nuclear social optimum: gas Bob Alice So, in state 1: expect that Alice will play top strategy Bob will play left strategy outcome is oil oil is social optimum Similarly, in state 2: gas is social optimum oil coal nuclear gas
• 18. Examples State 1 State 2 Alice Bob gas nuclear oil oil coal coal nuclear gas social optimum: oil Alice Bob nuclear oil gas gas coal coal oil nuclear social optimum: gas Bob Alice oil coal nuclear gas &#x2022; Thus, in either state, mechanism achieves social optimum, even though &#x2212; mechanism designer does not know the state herself &#x2212; Alice and Bob interested in own ends (not social goal) &#x2022; We say that mechanism implements the designer&#x2019;s goals (oil in state 1, gas in state 2)
• 19. Examples State 1 State 2 Alice Bob gas nuclear oil oil coal coal nuclear gas optimum: oil Alice Bob gas nuclear oil oil nuclear coal coal gas optimum: nuclear Let us change the example a bit: &#x2022; Wrongly set nuclear as social optimum, observe that although oil is optimal in state 1, it is not optimal in state 2, despite the fact that it falls in neither Alice&#x2019;s nor Bob&#x2019;s rankings between states 1 and 2 &#x2022; monotonicity is a property ensuring that the oil remain optimal in state 2 Theorem 1 (Maskin 1977): If a social choice rule is implementable, then it must be monotonic. Theorem 2 (Maskin 1977): Suppose that there are at least three individuals. If the social choice rule satisfies monotonicity and no veto power, then it is implementable.
• 20. Algorithmic Design &#x2022; Have shown you mechanisms in the cake, and energy examples &#x2022; What about that auction? &#x2022; Examples raise questions (among others): &#x2212; How can we implement such a mechanism? &#x2212; How does Computer Science come into play?
• 21. Algorithmic Settings &#x25CF; &#x25CF; &#x25CF; &#x25CF; An important part of Computer Science research deals with distributed settings These mainly focus on the connection of several computers and the computations these perform to achieve a common outcome according to some protocol (algorithm) It is generally assumed that the participants follow the instructions of the protocol This is obviously not always the case (e.g. the Internet)
• 22. Two example applications &#x25CF; Load Balancing: &#x2013; &#x2013; &#x25CF; In a &#x201C;perfect&#x201D; world, the aggregate computational power of all computers on the Internet would be optimally allocated online among connected processors, which is in itself a difficult problem In reality, all resources belong to individual entities that act in a rationally selfish way, which implies the necessity of some form of motivation for participation Routing: &#x2013; Information passes through several intermediate routers before reaching its intended destination &#x2013; Since routers are considered self-interested entities, this implies that the protocols employed should take the router's potential interests into consideration
• 23. Algorithmic Mechanism Design &#x25CF; &#x25CF; &#x25CF; &#x25CF; Nisan and Ronen [6] used notions of mechanism design to introduce a new framework for the study of such problems Theirs was not the first use of such notions in Computer Science studies, but arguably one of the most relevant and influential This marriage of concepts has even more interesting implications (e.g. complexity) We only briefly mention the first steps of what is now recognized as an important research direction
• 24. The model: Problems &#x25CF; Output specifications, defined algorithmically &#x2013; &#x2013; &#x25CF; Input is information about the setting (common) and the participating agents (their types &#x2013; private) Output is the specific computed outcome, based on the information above Descriptions of agents' desires (preferences) &#x2013; &#x2013; &#x25CF; A valuation function for every agent, based on its type and outcome A payment from the mechanism to every agent for participating, based on its type The optimization version has an objective function as an outcome
• 25. The model: Solutions &#x25CF; &#x25CF; &#x25CF; &#x25CF; A problem is solved when the required output is obtained, while agents try to achieve their goals (maximize their utilities) Agents have a specified family of strategies and the outcome depends on all these choices Depending on its strategy, the mechanism provides a payment to each agent, which can be used to make the agents' desires compatible with the required outcome Complexity matters
• 26. Desired characteristics of Mechanisms &#x25CF; Implementations with dominant strategies &#x2013; &#x2013; &#x25CF; Every agent has some dominant strategy Every set of dominant strategies yields a desired outcome Truthful Implementation &#x2013; &#x2013; &#x25CF; All agents want to report their type (no manipulation) Correctly reporting one's type is a dominant strategy The revelation principle states that given a mechanism that implements a problem with dominant strategies, there exists a truthful implementation as well
• 27. Vickrey-Groves-Clarke Mechanisms &#x25CF; &#x25CF; &#x25CF; &#x25CF; &#x25CF; Originally defined for auctions (second-price) VGC is a very useful type of mechanism, applied to maximization problems, where the objective is the sum of all agents' valuations Intuitively, the payment defined by VGC for an agent is its contribution to social welfare VGC mechanisms have been famously proven to be truthful (may even be the only truthful implementations) Complexity matters a lot
• 28. Shortest Paths Example &#x25CF; &#x25CF; &#x25CF; &#x25CF; &#x25CF; Consider a communication network, modeled by a directed graph There is one source and one sink node Every edge is an agent, whose type (private information) is te, the cost for sending a single message across this edge The goal is to find the cheapest path from the source to the sink (single message) Agent's valuation is 0 if its edge is not part of e the chosen path and -t otherwise
• 29. A Truthful Implementation &#x25CF; &#x25CF; When all agents honestly report their types (costs), the cheapest path can be calculated The following mechanism ensures the above is a dominant strategy for each agent: Set the payment for each agent to be pe=0 if its edge is not in the shortest path and pe=dG-e-dG|e=0 otherwise (according to inputs) &#x25CF; &#x25CF; This is a VGC mechanism, the computed paths are the shortest paths and truth-telling is a dominant strategy Similar ideas are being used in today's Internet
• 30. Conclusion &#x2022; Have seen some implemetations of the theory of Mechanism Design Many other potential applications &#x2022; &#x2013; &#x2013; &#x2013; &#x2022; &#x2022; Policies to prevent financial crises Sustainable gas emission policies Elections Design Combining Mechanism Design and Algorithmics is very natural and yields interesting and useful results Although we merely mentioned a small part of some introductory notions, the underlying theory is itself plentiful and well-founded
• 31. &#x25CF; &#x2022; Thank you! Questions?
• 32. References 1. 2. 3. 4. 5. 6. Yoav Shoham, Kevin Leyton-Brown Multiagent Systems Algorithmic, Game-Theoretic, and Logical Foundations, 2009 http://marginalrevolution.com/marginalrevolution/2007/10/mechanism-desig.html , Mechanism Design for Grandma by Alex Tabarrok on October 15, 2007 Matthew O. Jackson, Mechanism Theory , https://www2.bc.edu/~unver/teaching/gradmicro/mechtheo.pdf revised December 8 2003 Mechanism Design:How to Implement Social Goals (Eric S. Maskin) Mechanism Design:How to Implement Social Goals , presentation by Eric S. Maskin, Mechanism Design Theory(Harvard, Frankfurt, Shanghai, Prato, Berlin).pdf, stifterverband.info/.../maskin_mechanism_design_theory.pdf, visited 22-11-2012 Algorithmic Mechanism Design, Noam Nisan, Amir Ronen, Games and Economic Behavior, Volume 35, Issues 1-2, April 2001, Pages 166-196