Agent Technology for e-Commerce Chapter 10: Mechanism Design Maria Fasli

Chapter 10 Agent Technology for e-Commerce 2 The mechanism design problem  A set of N agents  Each agent has private information  i  i (its type) which determines its preferences over outcomes  Set of outcomes  An agent’s utility given its type  i is u i (o,  i )  Agents can reach an outcome by interacting through an institution or mechanism Problem  The outcome of the society depends on the agents’ types which are private information – the agents have to reveal them truthfully  How to implement an optimal outcome given private information

Chapter 10 Agent Technology for e-Commerce 3 Social choice function  A social choice function f :  1   …   n → selects the optimal outcome given the agents’ types and encapsulates the mechanism designer’s objectives  Pareto optimality  Maximization of total utility across agents

Chapter 10 Agent Technology for e-Commerce 4 Implementing social choice functions  A social choice function f can be indirectly implemented by having agents interact through an institution or mechanism  The mechanism design problem is then the problem of providing the ‘rules of the game’ to implement the solution to the social choice function when agents are self-interested and have types  i  A mechanism defines the strategies available; the rules of how the agent actions are turned into a social choice are given by the outcome function g(.)

Chapter 10 Agent Technology for e-Commerce 5 Given a mechanism M with outcome function g(.), M implements f(.), if the outcome computed with equilibrium agent strategies is a solution to the social choice function for all possible profiles of types  =(  1,…,  n )

Chapter 10 Agent Technology for e-Commerce 6 Dominant strategy implementation  What social choice functions can be implemented when the agents’ types (preferences) are private information?  Direct revelation mechanism: each agent is asked to reveal its type and given the type announcements the outcome rule selects an outcome  A strategy is truth-revealing if it reports true information about types  In an incentive compatible mechanism, the agents report their types truthfully

Chapter 10 Agent Technology for e-Commerce 7  A direct revelation mechanism where agents have a dominant strategy to reveal their true types is called strategy-proof or dominant strategy incentive compatible  Dominant strategy equilibrium of mechanism M: the agents play their dominant strategy  Mechanism M implements social choice function f(.) in dominant strategies, if there exists a dominant strategy equilibrium of M such that g(s*(  ))=f(  )

Chapter 10 Agent Technology for e-Commerce 8 Revelation principle Suppose that there exists a mechanism M that implements the social choice function f(.) in dominant strategies. Then f(.) is truthfully implementable in dominant strategies To identify which social functions are implementable in dominant strategies, we need only identify those functions f(.) for which truth-revelation is a dominant strategy for all agents in a direct revelation mechanism with outcome rule g(.)=f(.)

Chapter 10 Agent Technology for e-Commerce 9 Gibbard-Satterthwaite Impossibility Theorem If agents have general preferences and there are at least two agents, and at least three different optimal outcomes over the set of all agent preferences, then a social choice function is dominant strategy implementable if and only if it is dictatorial The results do not necessarily continue to hold in restricted environments

Chapter 10 Agent Technology for e-Commerce 10 Quasilinear environments  The agents’ utility takes the form u i (k,t i,  i ) = v i (k,  i )+t i  The outcome rule g(s) is decomposed into:  a choice rule k(s) which selects a choice given strategy profile s and  a payment rule t i (s) which selects a payment to agent i, based on strategy profile s

Chapter 10 Agent Technology for e-Commerce 11 The Groves mechanism  Given the reported preferences, the social choice rule in the Groves mechanism computes an optimal outcome as follows:  Choice k* maximizes the total reported value over all agents, i.e. it is ex post efficient  The payment rule in the Groves mechanism is then defined as:  Where h i is an arbitrary function on the reported types of every agent i

Chapter 10 Agent Technology for e-Commerce 12 The Clarke mechanism  Also known as the Pivotal or VCG mechanism is a special case of the Groves mechanism  Uses a taxing scheme: the amount an agent pays depends on how much it influences the outcome. In the Clarke mechanism: where is the optimal collective choice excluding i:

Chapter 10 Agent Technology for e-Commerce 13 Agent i’s transfer is then given by:  The payment rule h i (  -i ) is carefully set to achieve individual rationality  The Clarke mechanism is individual rational if the following conditions hold:  choice set monotonicity  normalization

Chapter 10 Agent Technology for e-Commerce 14 The Generalized Vickrey Auction  The GVA is an application of the Clarke mechanism to resource allocation problems  Suppose an auctioneer has a set of items X that it would like to allocate to a set of agents N on the condition: subject to the constraint: But the participants may not want to reveal their true valuations

Chapter 10 Agent Technology for e-Commerce 15  Each agent i reports a valuation function  The mechanism calculates the allocations and  Agent i receives bundle and receives a payment: The final payoff to the agent takes the form:

Chapter 10 Agent Technology for e-Commerce 16 Example: Vickrey auction  The valuation function of agent i is v i – p  x i =1 if agent i gets the item and x i =0 if it does not  The sum of the valuations is: and the resource constraint is

Chapter 10 Agent Technology for e-Commerce 17  Let m be the index of the agent with the maximum value of v i  To maximise the sum of valuations, the mechanism will allocate and x j = 0 for all j  m  Suppose l has the second-highest valuation  If agent m is eliminated, the maximum sum of the remaining valuations will be v l  The net payoff to agent m will be v m – v l, which is the result of the Vickrey auction

Chapter 10 Agent Technology for e-Commerce 18  Assume 2 agents and 3 units of a commodity to allocate  Agent A’s valuation (10, 8, 5)  Agent B’s valuation (9, 7, 6)  The optimal allocation is to give two units to A and one to B Using the GVA the problem is solved as follows:  If A is not present, all goods go to B with total value 9+7+6=22  A’s net payoff is 18+[9-22] = 5, so A pays 13 for the 2 units  If B is not present, all goods go to A with total value =23  B’s net payoff is then 9+[18-23]=4, so B pays 5 for one unit  The seller receives 13+5 for the three units sold

Chapter 10 Agent Technology for e-Commerce 19 Applications of the Clarke tax algorithm  Public project issue: to build or not a community gym  Residents decide by voting – those that vote yes, pay its cost  Some may decide to lie and once the gym is build to freeride it  u i (o)=v i (g)+  i where g=1 if gym is built, or g=0 if otherwise and  i is the numeraire (monetary transfer)  Agents declare their valuations, but may not be truthful

Chapter 10 Agent Technology for e-Commerce 20  Solution: make those agents whose vote changes the outcome, pay a tax. The social choice function is:  Every agent therefore has to pay a tax which is calculated as  The mechanism does not maintain budget balance as too much tax is collected

Chapter 10 Agent Technology for e-Commerce 21 Computational issues in MD Computation in mechanism design can be considered at two levels:  Agent level  Valuation complexity  Strategic complexity  At the infrastructure/mechanism level  Solution complexity  Communication complexity

Chapter 10 Agent Technology for e-Commerce 22  Mechanisms with dominant strategies are efficient giving them excellent strategic complexity  But the direct revelation property of Groves mechanisms provides very bad agent valuation complexity as the agent has to determine its complete preferences over all possible outcomes  The winner determination of Groves mechanisms in particular in combinatorial problems limits their applicability; CAP is NP-hard

Chapter 10 Agent Technology for e-Commerce 23 To resolve tension between game-theoretic and computational properties a number of approaches have been proposed:  Using approximation methods  Identifying tractable special cases within more general problems  Providing compact and expressive representation languages for agents to express their preferences  Employing dynamic instead of single-shot direct revelation mechanisms  Using decentralized mechanisms

Chapter 10 Agent Technology for e-Commerce 24 Decentralized mechanisms offer certain advantages  Tractability  Robustness  No trust needs to be placed on an entity which decides on the outcome  Communication bottlenecks can be avoided