Mechanism Design with Execution Uncertainty

Mechanism Design with

Execution Uncertainty

Ryan Porter, Amir Ronen,

Yoav Shoham, and Moshe Tennenholtz

Stanford University

August 3, 2002 Mechanism Design with Execution Uncertainty 2

Task Allocation

Motivating Example: Center = Government Task = Transport a package between two offices Agents = Shipping companies What protocol should the government use to allocate and pay for

the task? The general class of task allocation settings is well-studied, but

standard settings assume that the task is always completed.

task

CenterAgent 1

Agent N

value

cost

cost



What protocol should we use if there is uncertainty about the execution of the task?

First, we need to decide which type of execution uncertainty to consider.

We assume that agents make a full effort, but there is an inherent probability of failure.

We will revisit this assumption in the conclusion. Why Mechanism Design?

Consider the case of costless tasks, but firms only succeed with some probability.

What protocol should the government use?


Mechanism Design

Naïve Protocol: Firms declare probability to complete each task. Government assigns the task to the firm with the highest

declared probability for that task. Government pays this agent a fixed, positive amount if

succeeds, and nothing if it fails.

Problem: firms have incentive to lie about their probability of success.

This example shows the need for Mechanism Design (MD), the science of crafting protocols that give self-interested agents incentive to act in a way that achieves the goals of the center.


Outline

I. Single-Task Setting:

a. Formulation and mechanism goals

b. Mechanism that satisfies all goals

II. Multiple-Task Setting:

a. Different possible formulations

b. When dependencies exist:

1. Impossibility result

2. Possibility result for a relaxed set of goals

III. Conclusion


Formulation: Single-Task Setting

Single task t

N agents, each agent i is characterized by its privately-known

type θi2 θi = (pi,ci):

pi is the probability of completing t

ci is the cost of attempting t

Let θ = (θ1,…,θn)

Let θ-i = (θ1,…,θi-1,θi+1,…,θn). Then, θ = (θi,θ-i)

Center: has a value V for the completion of the task Necessary for making the correct allocation

V is common knowledge


Formulation: Mechanism

A mechanism consists of the following steps:

1. Nature reveals to each agent its type θi

2. Each agent declares θi Strategy for agent i is S: → (a direct mechanism)

Restriction allowed by the Revelation Principle

3. Center allocates task: A(θ) = ( A1(θ) , … , An(θ) )

4. Agents attempt their assignment (only one agent in this case) Result is recorded by the indicator variable µ

5. Center pays agents: R(θ,µ) = ( R1(θ,µ) , … , Rn(θ,µ) )

1. A mechanism is defined by the pair = (A(¢), R(¢))


Utility Functions

Utility function for agent i: ui(ci,θ,µ) = Ri(θ,µ) – ci(Ai(θ)

Agents are assumed to be expected-utility maximizers

Expectations always taken before the task is attempted

(that is, the expectation is over µ | p,A(θ))

Social welfare: W(A(θ),c,µ) = V· µ - i ci(Ai(θ))

Key idea of MD: align utility of each agent with social welfare


Mechanism Goals

Incentive Compatibility (IC): truth-telling (8 θi, si(θ i) = θi) is a

dominant strategy – 8 i,θ,θ-i,θi E[ui(ci,(θi,θ-i),µ)] ¸ E[ui(ci,(θi,θ-i),µ)]

Individual Rationality (IR): truthful agents always have nonnegative

expected utility – 8 i,θ,θ-i E[ui(ci,(θi,θ-i),µ)] ¸ 0

Center Rationality (CR): center’s utility always nonnegative – 8 θ,µ

V· µ - i Ri(θ,µ) ¸ 0

Social Efficiency (SE): allocation function maximizes expected social

welfare based on the true types –

8 θ, equilibria θ A(θ) = argmaxA' E[W(A',c,µ)]

Shows the use of IC, because A(·) can only depend on θ.

IC allows us to conclude that θ = θ.


Single-Task Setting: Only Costs

Second-Price Reverse Auction: Aj(θ) = {t} iff cj = c[1]

(That is, assign t1 to the agent with the lowest declared cost.)

Rj(θ,µ) = c[2]

(That is, pay this agent the second-lowest declared cost.)

agent cost (ci)

1 5

2 12

3 8

V = 10Example:


Single-Task Setting: Only Probabilities

Mechanism: Aj(θ) = {t} iff pj = p[1]

(Assign t1 to the agent with the highest declared probability.)

Rj(θ,µ) = – V·p[2] + V·µ

(Pay this agent according to the second-highest declared probability.)

Intuition: V·µ aligns the agent’s utility with social welfare, while (–

V·p[2]) gives us IC because no other agent could profit from R j(·).

agent probability (pi)

1 0.6

2 0.5

3 0.3

V = 10Example:


Single-Task Setting: General Case

Single-Task Mechanism: Aj(θ) = {t} iff j = argmaxi (pi ·V – ci)

(Assign the task to the agent i that maximizes expected welfare

E[W(A(θ),c,µ)] based on the declared types.)

Rj(θ,µ) = – E[W-j(A(θ-j),c,µ)] + V·µ

(Starting with a baseline loss of the expected welfare of the

system without this agent, agent j gets the value of the task

if it successfully completes the task.)

Theorem: Single-Task Mechanism satisfies IC, IR, CR, and SE.


Proof Sketch of IC

Plan: Derive an equation for the expected utility of a truthful agent, and then show it cannot do better.

Consider both cases for a truthful agent i: If agent i is assigned the task, then:

E[ui(ci,(θi,θ-i),µ)] = – E[W-i(A(θ-i),c,µ)] + pi·V – ci = – E[W-i(A(θ-i),c,µ)] +

E[W(A(θi,θ-i),c,µ)] If agent i is not assigned the task, then E[ui(·)] can be

written the exact same way. Agent i can only affect the second term, and the

mechanism optimizes this term assuming that agent i’s true type is θi.


Outline

I. Single-Task Setting:

a. Formulation and mechanism goals

b. Mechanism that satisfies all goals

Multiple-Task Setting:

a. Different possible formulations

b. When dependencies exist:

1. Impossibility result

2. Possibility result for a relaxed set of goals

III. Conclusion


Multiple-Task Setting

Simple Extension of the Single-Task Setting:

All variables become vectors of variables:

t = (t1,…,ts), pi = (p1i,…,ps

i), etc.

Since the tasks are independent (for both the center and the

agents), we can simply apply the Single-Task Mechanism to

each task separately.

Alternative formulations (skipped due to time constraints):

Combinatorial costs or probabilities for the agents

Combinatorial valuation function for the center


Multiple-Task Setting: Dependencies

Interesting settings have dependencies.

For example, object must be shipped from SF to Denver

before it can be shipped from Denver to Boston.

Definition: A task t2 is dependent on task t1 if t2 cannot be

attempted unless t1 is completed.

Dependencies induce a partial order in which the tasks are

attempted, and an agent does not incur the cost for an

assigned task that it cannot attempt.

t2

t1 t3

t4

t5

Nodes = tasks

Arcs = dependencies

(or, prerequisites)


Impossibility Result

Theorem: When dependencies exist between tasks, there

does not exist a mechanism that satisfies IC, IR, CR, and SE,

for any n,s ¸ 2.

Full proof is long, and it’s in the paper.

Intuition: Consider the case of: t1 assigned to agent 1 and t2 to agent 2.

How do you pay agent 2 in a way that satisfies IC?

A key part of the payment rule for the Single-Task Mechanism was the

ability to “test” the agent’s true probability of success.

However, agent 1 can block this test, and does so with a probability that is

unknown to the mechanism.

t2t1


Possibility Result

Relaxed set of goals: ex post Nash equilibrium versions of IC, IR, and SE.

No CR

The ex post Nash equilibrium: all agent truthfully reveal their type (8 i,θi si(θi) = θi). That is, even if each agent knew the true types of the other agents, truth-telling is still a best response.

Ex Post Nash Equilibrium Mechanism: A(θ) = A*(θ)

(Tasks are assigned according to A*(θ), the socially optimal allocation rule for the declared the types.)

8 i, Ri(θ,µ) = – E[W-i(A*(θ-i),c,µ)] + W-i(A*(θ),c,µ)

(Each agent i start with a loss equal to the expected social welfare

if this agent did not exist, and is then paid the actual welfare of the

entire system except for agent i.)


Conclusion

Main Idea:

Add the dimension of execution uncertainty to a basic

Mechanism Design problem for task allocation.

Main Results: Single-Task Setting:

We present a mechanism that satisfies all goals.

Multiple-Task Setting:

Dependencies prevent us from satisfying all goals, but

we present a mechanism that satisfies a relaxed set of

goals.


Future Work

Classes of execution uncertainty:This work:

1. Full effort, but inherent probability of failure

2. Rational, intentional failures

Future work:

3. “Irrational”, intentional failures

Mechanism Design with


Ryan Porter, Amir Ronen,

Yoav Shoham, and Moshe Tennenholtz

Stanford University


Introduction

Basic mechanism design problem for task allocation: Center has a set of tasks it wants completed

Agents have private costs to complete the tasks

Center constructs a protocol (mechanism) where: Agents pass information (e.g., bids) to the center

Center assigns tasks and makes payments

We add the dimension of uncertainty for the

completion of tasks


Dependencies

t2

t1 t3

t4

t5

Nodes = tasks

Arcs = dependencies

(or, prerequisites)


Single-Task Setting: Only Probabilities

Straw-man Mechanism:

Each agent declares a probability pi1

Assign the task to the agent with the high probability (p[1]1)

Pay this agent the difference in E[W-i(·)] (regardless of the

outcome of the attempt): (p[1]1 – p[2]

1) · V1

a1: p11 = 0.6

a2: p21 = 0.5

a3: p31 = 0.3

t1

V1 = 10

Example:


Impossibility Result: Proof Sketch

SE implies: A1(θ) = {t2} and A2(θ) = {t1}

Three variables to set: R1(θ , ;) , R1(θ , {t1}) and R1(θ , {t1,t2})

R1(θ , ;) = 0 – otherwise, IR violated or a1 would lie if p12 = 0

So, set up “sequential” mechanisms, where the second is only

implemented if t1 is completed: R1(θ , {t1,t2}) = 9 (if a1 succeeds)

R1(θ , {t1}) = -1 (if a1 fails)

This lack of IC because of the dependency is unavoidable

a1: p11 = 0.4 p1

2 = 1.0

a2: p21 = 0.0 p2

2 = 0.1

t1 t2

V1 , V2 = 10

8 i,s cis = 0

p21 = 0.5

{

p11 = 0.6


agent 1: c11 = 5

agent 2: c12 = 12

agent 3: c13 = 8

V1 = 10Example:t1

Mechanism Design with Execution Uncertainty

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