Social Choice & Mechanism Design

Transcription

1 Decision Making in Robots and Autonomous Agents Social Choice & Mechanism Design Subramanian Ramamoorthy School of Informatics 2 April, 2013

2 Introduction Social Choice Our setting: a set of outcomes agents have preferences across them for the moment, we won't consider incentive issues: center knows agents' preferences, or they declare truthfully the goal: a social choice function: a mapping from everyone's preferences to a particular outcome, which is enforced how to pick such functions with desirable properties? 2/4/2013 2

3 Formal Model 2/4/2013 3

4 Non-ranking Voting Schemes Plurality pick the outcome which is preferred by the most people Cumulative voting distribute, e.g., 5 votes each possible to vote for the same outcome multiple times Approval voting accept as many outcomes as you like" 2/4/2013 4

5 Ranking Voting Schemes Plurality with elimination ( instant runoff ) everyone selects their favorite outcome the outcome with the fewest votes is eliminated repeat until one outcome remains Borda assign each outcome a number. The most preferred outcome gets a score of n-1, the next most preferred gets n-2, down to the nth outcome which gets 0. Then sum the numbers for each outcome, and choose the one that has the highest score Pairwise elimination in advance, decide schedule for the order in which pairs will be compared given two outcomes, have everyone determine the one that they prefer eliminate the outcome that wasn t preferred, continue with the schedule 2/4/2013 5

6 Condorcet Condition If there is a candidate who is preferred to every other candidate in pairwise runoffs, that candidate should be the winner While the Condorcet condition is considered an important property for a voting system to satisfy, there is not always a Condorcet winner Sometimes, there is a cycle where A defeats B, B defeats C, and C defeats A in their pairwise runoffs 2/4/2013 6

7 Voting Paradox: Condorcet Example What is the Condorcet winner? B What would win under plurality voting? A What would win under plurality with elimination? C 2/4/2013 7

8 Voting Paradox: Sensitivity to Losing Candidate What candidate wins under plurality voting? A What candidate wins under Borda voting? A Consider dropping C. What happens under Borda & Plurality? B wins. 2/4/2013 8

9 Voting Paradox: Sensitivity to Agenda Setter Who wins pairwise elimination, with the ordering A,B,C? C Who wins with the ordering A,C,B? B Who wins with the ordering B,C,A? A 2/4/2013 9

10 Desirable Properties 2/4/

11 Property: Pareto Efficiency 2/4/

12 Property: Independence of Irrelevant Attributes 2/4/

13 Property: Non-dictatorship 2/4/

14 Arrow s Theorem Assume that W is both PE and IIA Then, one can show that W must be dictatorial Our Assumption that O is at least 3 is necessary for the proof 2/4/

15 On Social Choice Functions Perhaps the issue with Arrow's theorem is really that we require a whole preference ordering. Is it the case that a social choice function might be easier to find? One needs to redefine criteria for the social choice function setting; PE and IIA discussed the ordering 2/4/

16 Weak Pareto Efficiency 2/4/

17 Monotonicity 2/4/

18 Non-dictatorship 2/4/

19 Result 2/4/

20 Social Choice Bayesian Game 2/4/

21 Mechanism Design 2/4/

22 The Problem Pick a mechanism that causes rational agents to behave in a desired way; maximizing the mechanism designer's own utility or objective function each agent holds private information, in the Bayesian game sense often, we're interested in settings where agents' action space is identical to their type space, and an action can be interpreted as a declaration of the agent's type Various equivalent ways of looking at this setting solve an optimization problem, given that the values of (some of) the inputs are unknown choose the Bayesian game out of a set of possible Bayesian games that maximizes some performance measure design a game that implements a particular social choice function in equilibrium, given that the designer no longer knows agents' preferences and the agents might lie 2/4/

23 Implementation in Dominant Strategies 2/4/

24 Implementation in Bayes-Nash Equilibrium 2/4/

25 BNE Implementation Bayes-Nash Equilibrium Problems: there could be more than one equilibrium which one should we expect agents to play? agents could miscoordinate and play none of the equilibria asymmetric equilibria are implausible Refinements: Symmetric Bayes-Nash implementation Ex-post implementation 2/4/

26 Implementation Comments We can require that the desired outcome arises in the only equilibrium in every equilibrium in at least one equilibrium Forms of implementation: Direct Implementation: agents each simultaneously send a single message to the center Indirect Implementation: agents may send a sequence of messages; in between, information may be (partially) revealed about the messages that were sent previously like extensive form 2/4/

27 Revelation Principle Any social choice function that can be implemented by any mechanism can be implemented by a truthful, direct mechanism Consider arbitrary, non-truthful mechanism (maybe indirect) Recall that a mechanism defines a game, and consider an equilibrium s = (s 1,, s n ) 2/4/

28 Revelation Principle We can construct a new direct mechanism, as shown above This mechanism is truthful by exactly the same argument that s was an equilibrium in the original mechanism The agents don't have to lie, because the mechanism already lies for them 2/4/

29 An Impossibility Result 2/4/

30 Implications We should be discouraged about the possibility of implementing arbitrary social-choice functions in mechanisms. However, in practice we can circumvent the Gibbard- Satterthwaite theorem in two ways: use a weaker form of implementation note: the result only holds for dominant strategy implementation, not e.g., Bayes-Nash implementation relax the onto condition and the (implicit) assumption that agents are allowed to hold arbitrary preferences 2/4/

31 Example of a Mechanism: VCG Auction Vickrey Clarke Groves auction of multiple goods is a sealedbid auction where bidders report their valuations The auction system assigns items in a socially optimal manner This system charges each individual the 'harm' they cause to other bidders, and ensures that the optimal strategy for a bidder is to bid the true valuations of the objects. The auction is named after William Vickrey, Edward H. Clarke and Theodore Groves who successively generalized the idea 2/4/

32 VCG Auction There is a set M of auctioned items and a set N of bidders is the social value for a given bid combination A bidder b i who wins an item t j pays: This is the social cost of his winning incurred by the rest The winning bidder who has value A derives utility: 2/4/

33 Why is this a Good Idea? Suppose b 1 wins t 1 upon submitting his true valuation Bid size of b 1 has no effect on his utility as long as he wins the item. Let us assume he is not truthful, receiving t j Utility in truthful revelation case: Utility in the un-truthful revelation case: The difference: Must be higher because it was allocated by VCG 2/4/

34 Concluding Remarks The social choice problem may be thought of in terms of how preference orders are mapped to a global choice of outcome or a global order There are many desirable conditions but these are not always easy to attain Mechanism design is a way of constructing games in order to implement social choice functions This still suffers from similar impossibility problems but under suitable conditions (in practice) it can be done VCG a famous example in the auction setting 2/4/

35 Acknowledgements These slides are adapted from the lecture notes of Kevin Leyton-Brown 2/4/