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

Agent Technology for e-Commerce

Chapter 10: Mechanism Design

Maria Faslihttp://cswww.essex.ac.uk/staff/mfasli/ATe-Commerce.htm

2Chapter 10


The mechanism design problem

A set of N agents Each agent has private information ii (its type) which

determines its preferences over outcomes Set of outcomes An agent’s utility given its type i is ui(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

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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

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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(.)

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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)

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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

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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()

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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(.)

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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

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Quasilinear environments

The agents’ utility takes the form ui(k,ti,i) = vi(k,i)+ti

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 ti(s) which selects a payment to agent i, based

on strategy profile s

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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 hi is an arbitrary function on the reported types of every agent i

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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:

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Agent i’s transfer is then given by:

The payment rule hi(-i) is carefully set to achieve individual rationality

The Clarke mechanism is individual rational if the following conditions hold:

choice set monotonicity normalization

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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

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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:

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Example: Vickrey auction

The valuation function of agent i is vi – p

xi=1 if agent i gets the item and xi=0 if it does not The sum of the valuations is:

and the resource constraint is

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Let m be the index of the agent with the maximum value of vi

To maximise the sum of valuations, the mechanism will allocate

and xj = 0 for all jm Suppose l has the second-highest valuation If agent m is eliminated, the maximum sum of the remaining

valuations will be vl

The net payoff to agent m will be vm – vl , which is the result of the Vickrey auction

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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 10+8+5=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

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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 ui(o)=vi(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

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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

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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

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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

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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

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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

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

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