Insincere Agents

1
Insincere Agents

• The agents chose the best strategy for them.
• A protocol implements a specific social choice
  function if the protocol has an equilibrium that
  results in the same outcome even when insincere
  agents do not truthfully reveal their
  preferences.

2
Mechanism Design

• Mechanism design is a strategic version of social
  choice theory.
• Mechanisms designed assuming that agents attempt
  to maximize their individual payoff.
• May not vote truthfully.
• The designer determines the desired behavior and
  then determines what strategic interaction will
  result in the behaviors.
• Closely related to designing effective protocols
  for distributed systems.

3
Bayesian Games

• A Bayesian game is a tuple (N, O, Q, p, u) where
• N is a set of agents,
• O is the set of outcomes,
• Q Q1 x x Qn is a set of possible joint type
  vectors.
• p Q 0, 1 is the common prior over types,
• u (u1, . . . , u1), where u1 O Q R is
  the utility function for player i.

4
Mechanism Design

• A mechanism (over a set of agents N and a set of
  outcomes O) is a pair (A, M), where
• A A1 An, where Ai is the set of
  actions available to agent i N, and
• M A P(O) maps each action profile to a
  distribution over outcomes.
• A mechanism is deterministic if for every action
  there exists an outcome such that M(a)(o) 1.
• The designer specifies
• The action sets for the agents (though they may
  be constrained by the environment)
• The mapping to outcomes, over which agents have
  utility.
• Cannot change the agents preferences for
  outcomes or type spaces.

5
Mechanism Design

• The problem is to choose a mechanism that causes
  rational agents to behave in a particular way, in
  order to maximize the mechanism designers own
  utility or objective function.
• Each agent has private information, in the
  Bayesian game sense.
• Often focused on settings where agents action
  space is identical to their type space, and an
  action can be interpreted as a declaration of the
  agents type.
• Various equivalent ways of looking at this
  setting
• Treat it as 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

6
Dominant Strategies

• A mechanism (A,M) (over N and O) is an
  implementation in dominant strategies of a social
  choice function C over N and O if for any vector
  of utility functions u, the game has an
  equilibrium in dominant strategies, and in any
  such equilibrium a we have M(a) C(u).

This is a key property whether it induces
outcomes that are consistent With a given social
choice function. A mechanism that results in
dominate strategies is sometimes
called Strategy-proof since agents do not need to
reason about each others Actions in order to
maximize utility.
7
Bayes-Nash Implementation

• Given a Bayesian game, a mechanism (A,M) is
  Bayes-Nash implementation of a social choice
  function C if there exists a Bayes-Nash
  equilibrium of the game of incomplete information
  (N, A, Q, p, u) such that for every q Q and
  every action profile a A that can arise given
  type profile q in this equilibrium, we have that
  M(a) C(u(, q)).

Auction design is a good example of Bayesian Mech
design.
8
Bayes-Nash Implementation

• Problems with the Bayes-Nash Equilibrium
• It is possible that there is more than one
  equilibrium.
• Which equilibrium should the agents play?
• Agents could miss-coordinate and play none of the
  equilibria.
• Asymmetric equilibria are implausible.
• Refinements
• Symmetric Bayes-Nash implementation
• Ex-post Bayes-Nash implementation

9
Mechanism Implementation

• It can be required that the desired outcome
  arises
• in the only equilibrium
• in every equilibrium
• in at least one equilibrium

10
Mechanism Implementation

• The mechanism may also be required to satisfy
  properties such as
• Individual rationality Agents are better off
  playing than not playing. No agent is harmed by
  participating.
• Budget balance The mechanism gives away and
  collects the same amounts of money. Pays out and
  receives the same amount of money.
• Truthfulness Agents honestly report their types
  or preferences to the mechanism.

11
Mechanism Implementation

• Forms of implementation
• Direct Implementation agents each simultaneously
  send a single message truthfully revealing their
  preferences 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.

12
Revelation Principle

• Revelation Principle Theorem
• If there exists any mechanism that implements a
  social choice function C in dominate strategies
  then there exists a direct mechanism that
  implements the C in a dominate strategies and is
  truthful.
• Any solution to a mechanism design can be
  converted to one in which the agents always
  reveal their true preferences, if the new
  mechanism lies in the same way the agents would
  have lied to the original mechanism.

13
Revelation Principle

• This may not be possible for computationally
  limited agents.
• The version of the theorem using the Nash
  equilibrium has issues
• It assumes that the agents and the designer have
  common knowledge of the joint probabilities of
  the agents types.
• The revised protocol may have multiple Nash
  equilibria and the theorem assumes only one
  exists.

14
Revelation Principle

• Truthfulness can always be achieved!
• Consider an arbitrary, non-truthful mechanism
  (e.g., may be indirect).
• Recall that a mechanism defines a game, and
  consider an equilibrium s (s1, . . . , sn).

Any solution to a mech design prob can be
converted into one in which Agents always reveal
their true preferences, if the mech lies for them
in the Same way they would have lied with the
original mech.
15
Revelation Principle

• A new direct mechanism can be constructed.
• This mechanism is truthful by exactly the same
  argument that s was an equilibrium in the
  original mechanism
• The agents dont have to lie, because the
  mechanism already lies for them.

16
Revelation Principle

• A computational criticism of the revelation
  principle is that computation is pushed onto the
  center.
• The agents strategies will often be
  computationally expensive.
• e.g., in the shortest path problem, agents may
  need to compute shortest paths, cutsets in the
  graph, etc.
• Since the center plays equilibrium strategies for
  the agents, the center now incurs this cost.
• If computation is intractable, so that it cannot
  be performed by agents, then in a sense the
  revelation principle does not hold.
• The agents cannot play the equilibrium strategy
  in the original mechanism.
• However, it is unclear what the agents will do.

17
Revelation Principle

• The set of equilibria is not always the same in
  the original mechanism and the revelation
  mechanism.
• Of course, it is shown that the revelation
  mechanism does have the original equilibrium of
  interest.
• However, in the case of indirect mechanisms, even
  if the indirect mechanism had a unique
  equilibrium, the revelation mechanism can also
  have new, bad equilibria.
• What is the revelation principle good for?
• Recognition that truthfulness is not a
  restrictive assumption.
• For analysis purposes, one can consider only
  truthful mechanisms, and be assured that such a
  mechanism exists.
• Recognition that indirect mechanisms cannot do
  (inherently) better than direct mechanisms.

Use the revelation principle carefully!
18
Dominant Strategy Implementation

• What social choice functions can be implemented
  in dominant strategies?
• Focus on direct mechanisms where strategy space
  is the space of agent preferences. Or more
  simply, the social choice function over the
  revealed preferences. So which social choice
  functions are truthful?

19
Impossibility Result

• Theorem (Gibbard-Satterthwaite) Consider any
  social choice function C of N and O. If
• O 3
• C is onto that is, for every o O there is a
  preference vector such that C( ) o (this
  property is sometimes also called citizen
  sovereignty) and
• C is dominant-strategy truthful,
• then C is dictatorial.

This negative result is specific to the
dominant-strategy implementation. Does not hold
for Nash, Bayes-Nash, or ex-post Bayes-Nash
Implem. Essentially saying, non-manipulable
protocols are dictatorial.
20
Impossibility Result

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

21
Quasilinear Utility

• Agents have quasilinear preferences in an
  n-player Bayesian game when the set of outcomes
  is O X Rn for a finite set X, and the utility
  of an agent i with type q is given by ui(o, q)
  ui(x, q) - fi(pi), where o (x, pi) is an
  element of O, ui X x Q R is an arbitrary
  function and fi R R is a strictly monotonically
  increasing function.

A quasilinear environment requires 1. No agent
cares how the other agents divide the payoffs
amongst themselves. 2. An agents gross benefit
should not depend on the amount of money the
agent will have. X represents the finite set of
non-monetary outcomes, allocating items to
bidders. pi is the (possibly negative) payment
made by the agent i. the mechanism can be thought
of as the auctioneer.
22
Quasilinear Utility

• Split the mechanism into a choice rule and a
  payment rule
• x X is a discrete, non-monetary outcome
• pi R is a monetary payment (possibly negative)
  that agent i must make to the mechanism.
• Implications
• ui(x, q) is not influenced by the amount of money
  an agent has.
• Agents do not care how much others are made to
  pay. (though they can care about how the choice
  affects others.)

23
Risk Attitudes

• How much is 1 worth?
• What are the units in which this question should
  be answered?
• Utils (units of utility)
• Different amounts depending on the amount of
  money you already have.
• How much is a gamble with an expected value of 1
  worth?
• Possibly different amounts, depending on how
  risky it is.
• So, what is fi(pi)?

24
Risk Neutrality
Minimize expected revenue. Indifferent to
participating or not therefore A linear
function Curvature of fi is the risk attitude.
25
Risk Aversion
A sublinear value for utility, prefers a sure
thing.
26
Risk Seeking
Superlinear value function, prefers to
participate in lottery rather Than a sure thing
with an expected value. People/agents may exhibit
different risk attitudes at different regions Of
fi.
27
Risk Seeking

• Assume that the agents are Risk Neutral.
• Risk Neutral has a linear slope of fi(pi), thus
  the agents have transferable utility.
• Regardless of the nonmonetary choice x X, one
  agent can transfer any given amount of utility to
  another agent be giving the appropriate amount of
  money.

28
Quasilinear Mechanisms

• A mechanism in the quasilinear setting (over a
  set of agents N and a set of outcomes O X Rn)
  is a triple (A, c , p), where
• A A1 An, where Ai is the set of
  actions available to agent i N,
• c A P(X) maps each action profile to a
  distribution over choices, and
• p A Rn maps each action profile to a payment
  for each agent.

Modify the definition Quasilinear preferences
split the outcome space into two parts. Split
the Function M into two functions c and p, where
c is the choice rule and p is the payment rule.
29
Quasilinear Mechanisms

• A direct quasilinear mechanism (over a set of
  agents N and a set of outcomes O X Rn) is a
  pair (c, p). It defines a standard mechanism in
  the quasilinear setting, where for each i, Ai
  Qi.
• An agents valuation for choice x X vi(x)
  ui(x, q)
• the maximum amount i would be willing to pay to
  get x.
• in fact, i would be indifferent between keeping
  the money and getting x.
• Equivalent definition mechanisms that ask agents
  i to declare vi(x) for each x X.
• Define vi as the valuation that agent i declares
  to such a direct mechanism.
• may be different from the agents true valuation
  vi.
• Also define the tuples v, v-i.

30
Truthfulness

• A quasilinear mechanism is truthful if
  , agent is equilibrium strategy is to adopt
  the strategy vi vi.
• Our definition before, adapted for the
  quasilinear setting equivalent definition of
  truthfullness

31
Efficiency

• A quasilinear mechanism is efficient is strictly
  Pareto efficient, of simply efficient, if it
  selects a choice x such that
• An efficient mechanism selects the choice that
  maximizes the sum of agents utilities,
  disregarding monetary payments agents must make.
• Called economic efficiency.
• Also called social-welfare maximization.
• Note defined in terms of true valuations, not
  declared valuations.

32
Budget Balance

• A quasilinear mechanism is budget balanced when
• where s is the equilibrium strategy profile.
• Regardless of the agents types, the mechanism
  collects and disburses the same amount of money
  from and to the agents.
• Relaxed version of weak budget balance
• .
• The mechanism never takes a loss, but it may make
  a profit.
• Budget balance can be required to hold ex ante
• .
• The mechanism must break even or make a profit
  only on expectation.

33
Individual Rationality

• A quasilinear mechanism is ex-interim individual
  rational when
• where s is the equilibrium strategy profile.
• No agent loses by participating in the mechanism.
• Ex-interim because it holds for every possible
  valuation for agent i, but averages over the
  possible valuations of the other agents.
• A quasilinear mechanism is ex-post individual
  rational when
  , where s is the equilibrium strategy
  profile.

34
Tractability

• A quasilinear mechanism is tractable when
  and can be computed in polynomial time.
• The mechanism is computationally feasible.

35
Revenue Maximization

• A quasilinear mechanism is revenue maximizing
  when, among the set of functions c and p which
  satisfy the other constraints, the mechanism
  selects the c and p that maximize
  ,where s(v) denotes the agents equilibrium
  strategy.
• The mechanism designer choose among mechanisms
  that satisfy the desired constraints by adding an
  objective function, such as revenue maximization.

36
Groves Mechanisms

• Recall that in the quasilinear utility setting, a
  mechanism can be defined as a choice rule and a
  payment rule.
• The Groves mechanism is a mechanism that
  satisfies
• dominant strategy (truthful implementation of a
  social-welfare maximizing sociela choice
  function.)
• Efficiency Considered one of the most important
  properties for mechanism design for QL
  mechansims.
• In general Groves Mechanism is not
• budget balanced
• individual-rational
• However, can recover these properties.

37
Groves Mechanisms

• The Groves mechanism is a direct quasilinear
  mechanism where,

Are a direct mech in which agents can declare any
valution function v The mech then optimizes the
choice of outcome assuming that the Agents
disclosed their true utility function.
38
Groves Mechanisms

• The choice rule should not come as a surprise
  since the mechanism is both truthful and
  efficient these properties entail the given
  choice rule.
• So what is going on with the payment rule?
• the agent i must pay some amount that
  does not depend on his own declared valuation.
• the agent i is paid , the sum of the
  others
• valuations for the chosen outcome.

39
Groves Truthfulness

• Theorem Truth telling is a dominant strategy
  under the Groves mechanism.

Groves mechs provide a dominant-strategy truthful
implementation of A social welfare maximizing
social choice function. It is easy to see That if
a Groves mech is dominant strategy truthful, then
it must be social-welfare maximizing. Groves
mech are dominant-strategy truthful because
agents Externalities are internalized. However,
an agents utility does not Depend only on the
selected choice, because payments are
imposed. Agents are paid the (reported) utility
of all other agents under the Chosen allocation,
each agent becomes just as interested
in Maximizing the other agents utilities as in
maximizing his own. Groves mech are only mechs
that implement an efficient allocation
in Dominant strategies among agents with
arbitrary quasilienar utilities.
40
Groves Uniqueness

• Theorem An efficient social choice function
  can be implemented in dominant
  strategies for agents with unrestricted
  quasilinear utilities only if
  
• This same result also holds for the broader class
  of Bayes-Nash incentive-compatible efficient
  mechanisms.

41
VCG Mechanism Clarke Tax

• The Clarke tax sets the hi term in a Groves
  mechanism as
• where c is the Groves mechanism allocation
  function.
• Theorem If each agent has quasilinear
  preferences, then using the Clarke Tax algorithm,
  each agents dominant strategy is to reveal his
  true preferences.

• algorithm leads to the most socially preferred
  outcome to be chosen.
• Truthtelling is every agents dominant strategy.
• Reduces speculation regarding others.
• The Clarke Tax algorithm does not maintain budget
  balance, too much tax can be collected.
• Other algorithms may not collect enough tax.
• This algorithm is not Pareto efficient because
  the surplus cannot be returned to the agents or
  anything that the agents care about because it
  will affect the agents utility and truthtelling.
  
• This algorithm also permits agents to develop
  coalitions that can affect the outcome.

42
VCG Mechanism

• The Vickrey-Clarke-Groves (VCG) mechanism is a
  direct quasilinear mechanism where
• You get paid everyones utility under the
  allocation that is actually chosen except your
  own, but you get that directly as utility.
• Then you get charged everyones utility in the
  world where you do not participate.
• Thus you pay your social cost.

43
VCG Mechanism

• who pays 0?
• agents who dont affect the outcome
• who pays more than 0?
• (pivotal) agents who make things worse for others
  by existing
• who gets paid?
• (pivotal) agents who make things better for
  others by existing
• Because only pivotal agents have to pay, VCG is
  also called the pivot mechanism.
• It is dominant strategy truthful, because it is a
  Groves mechanism

44
VCG Selfish routing

• What outcome will be selected by c? path ABEF.
• How much will AC have to pay?
• The shortest path taking his declaration into
  account has a length of 5, and imposes a cost of
  -5 on agents other than him (because it does not
  involve him). Likewise, the shortest path without
  ACs declaration also has a length of 5. Thus,
  his payment pAC (-5) - (-5) 0.
• This is what we expect, since AC is not pivotal.
• Likewise, BD, CE, CF and DF will all pay zero.

45
VCG Selfish routing

• How much will AB pay?
• The shortest path taking ABs declaration into
  account has a length of 5, and imposes a cost of
  2 on other agents.
• The shortest path without AB is ACEF, which has a
  cost of 6.
• Thus pAB (-6) - (-2) -4.

46
VCG Selfish routing

• How much will BE pay? pBE (-6) - (-4) -2.
• How much will EF pay? pEF (-7) - (-4) -3.
• EF and BE have the same costs but are paid
  different amounts. Why?
• EF has more market power for the other agents,
  the situation without EF is worse than the
  situation without BE.

47
VCG Individual Rationality

• An environment exhibits choice-set monotonicity
  if , X-i X.
• Removing any agent weakly decreasesthat is,
  never increasesthe mechanisms set of possible
  choices X.
• An environment exhibits no negative externalities
  if .
• Every agent has zero or positive utility for any
  choice that can be made without his
  participation.

48
Example Simple Exchange

• Consider a market setting consisting of agents
  interested in buying a single unit of a good such
  as a share of stock, and another set of agents
  interested in selling a single unit of this good.
  The choices in this environment are sets of
  buyer-seller pairings (prices are imposed through
  the payment function).
• If a new agent is introduced into the market, no
  previously-existing pairings become infeasible,
  but new ones become possible thus choice-set
  monotonicity is satisfied.
• Because agents have zero utility both for choices
  that involve trades between other agents and no
  trades at all, there are no negative
  externalities.

49
VCG Individual Rationality

• Theorem The VCG mechanism is ex-post individual
  rational when the choice set monotonicity and no
  negative externalities properties hold.

50
VCG Budget Balance

• An environment exhibits no single-agent effect if
  there exists a
  choice x that is feasible without i and that has
  
• Removing an agent does not worsen the total value
  of the best solution to the others, regardless of
  their valuations.
• Example Consider a single-sided auction.
  Dropping an agent just reduces the amount of
  competition, making the others better off.
  Removing an agent does not worsen the total value
  of the best solution to the other, regardless of
  their valuations.

51
VCG Budget Balance

• Theorem The VCG mechanism is weakly
  budget-balanced when the no single-agent effect
  property holds.
• Good news

52
VCG Balance Budget

• The bad news
• Theorem No dominant strategy incentive-compatible
  mechanism is always both efficient and weakly
  budget balanced, even if agents are restricted to
  the simple exchange setting.
• Theorem No Bayes-Nash incentive-compatible
  mechanism is always simultaneously efficient,
  weakly budget balanced and ex-interim individual
  rational, even if agents are restricted to
  quasilinear utility functions.

53
VCG Caveats

• VCG can end up paying arbitrarily more than an
  agent is willing to accept (or equivalently
  charging arbitrarily less than an agent is
  willing to pay)
• Consider AC, which is not part of the shortest
  path.
• If the cost of this edge increased to 8, our
  payment to AB would increase to pAB (-12) -
  (-2) -10.
• If the cost were any x 2, we would select the
  path ABEF and would have to make a payment to AB
  of pAB (-4 - x) - (-2) -(x 2).
• The gap between agents true costs and the
  payments that they could receive under VCG is
  unbounded.

54
VCG Caveats Privacy

• VCG requires agents to fully reveal their private
  information that may have value to agents
  extending beyond the current interaction.
• For example, the agents may know that they will
  compete with each other again in the future.
• It is often preferable to elicit only as much
  information from agents as is required to
  determine the social welfare maximizing outcome
  and compute the VCG payments.

55
VCG Caveats Collusion

• What happens if agents 1 and 2 both increase
  their declared valuations by 50?
• The outcome is unchanged, but both of their
  payments are reduced.
• Thus, while no agent can gain by changing his
  declaration, groups can.

56
VCG Caveats Returning Profits

• One may want to use VCG to induce agents to
  report their valuations honestly, but may not
  want to make a profit by collecting money from
  the agents.
• May want to find some way of returning the
  mechanisms profits back the agents.
• The possibility of receiving a rebate after the
  mechanism has been run changes the agents
  incentives.
• Even if profits are given to a charity that the
  agents care about, or spent in a way that
  benefits the local economy and hence benefits the
  agents, the VCG mechanism is undermined.
• Thus, burning the money collected by the
  mechanism is the only way ensuring that the
  agents incentives are not altered!

57
VCG Caveats

• VCG is not frugal. Can end up paying more than
  the lowest cost.
• VCG can result in less revenue when more agents
  are included. Revenue monotonicity a
  mechanisms revenue always weakly increases as
  agents are added to the mechanism.
• VCG may not satisfy the tractability property.
  NP-Hard.

58
AGV Mechanism

• The AGV mechanism is an alternative to VCG that
• Removes the ex interim individual rationality and
  dominant strategy requirements.
• Adds budget balance and ex ante individual
  rationality requirements.

59
AGV Mechanism

• The Arrow dAspremont-Gérard-Varet (AGV)
  mechanism is a direct quasilinear mechanism (c,
  p), where

ESW expected social welfare. AGVs allocation
rule is the same as Groves mechanism. AGV is
incentive compatible, so it is efficient.
Last Line guarantees a balanced budget. Requires
two sacrifices AGV is Truthful only in
Bayes-Nash equilibrium, and is only ex ante
individually Rational.
60
AGV Tractability

• AGV fails to satisfy the tractability property.
• Address this issue by
• Developing dominant-strategy mechanisms that
  implement different social choice functions.
• Build mechanism that use a Groves payment rule
  with an alternative choice function.

61
AGV Dominant Strategies

• Assume deterministic mechanisms.
• Thm A direct, deterministic mechanism is
  dominant-strategy incentive-compatible iff, for
  every i N and every
• The payment function can be written as
  and
• For every

First condition says that an agents payment can
only depend on other Agents declarations and
the selected choice, and not on the agents
own Declaration. Second condition says that
taking the other agents declarations and
the Payment function into account, from every
agents perspective the Mechanism selects the
most preferable choice.
62
AGV Dominant Strategies

• AGV creates a strong link between choice rules
  and payment rules.
• Can we characterize choice rules that work with
  dominant strategies, but do not reference
  payments?

63
AGV Dominant Strategies

• A social choice function C satisfies weak
  monotonicity (WMON) if for all i N and all
  implies that
• Thm All social choice functions implementable by
  deterministic dominant-strategy
  incentive-compatible mechanisms in quasilinear
  settings satisfy WMON. Furthermore, let C be an
  arbitrary social choice function C V1 x X Vn ?
  X satisfying WMON and having the property that
  , Vi is a convex set. Then C can be
  implemented in dominant strategies.

WMON says that any time the choice functions
decisions can be altered By a single agent
changing his declaration, it must be the case
that This change expressed a relative increase
in preference for the new Choice over the old
choice.
The convexity restriction is often acceptable.
64
AGV Dominant Strategies

• WMON is a local characterization that treats each
  agent individually.
• Really need a global characterization.
• A social choice function is an affine maximizer
  if it has the form
• where each gx is an arbitrary constant (perhaps
  -8) and each .

Affine maximizers are the only social choice
functions implementable with Dominant strategies.
65
AGV Dominant Strategies

• Thm If there are at least three choices that a
  social choice function will select given some
  input, and if agents have general quasilinear
  preferences, then the set of (deterministic)
  social choice functions implementable in dominant
  strategies is precisely the set of affine
  maximizers.

Affine maximizers are a weighted Groves mechanism
and transform Both choices and the agents
valuations by applying linear weights, then
Effectively run a Groves mechanism on the
transformed space. Thus far we have assumed a
quasilinear game situation. This assumption does
not hold for all domains.
66
AGV Tractable Groves Mechanisms

• An alternative is to develop a tractable Groves
  Mechanism.
• Requires inefficient social choice functions.
• Groves-based mechanisms are direct quasilinear
  mechanisms (c, p) for which
• is an arbitrary function mapping type
  declarations to choices, and

This mechanism uses Groves payment function,
regardless of what Allocation function it uses.
It is not dominant-strategy truthful. Only way an
agent can gain by lying to a Groves-based
mechanism is to Help it by causing it to select a
more efficient allocation. Do this with The
second chance mechanism.
67
AGV Tractable Groves Mechanisms

• Given a Groves-based mechanism (c, p), a
  second-chance mechanism works as follows
• Each agent i is asked to submit a valuation
  declaration and an appeal function l V?
  V.
• The mechanism computes , and also for
  all i N. From the set of choices thus
  identified, the mechanism keeps one that
  maximizes the sum of agents declared valuations.
  
• The mechanism charges each agent i

An appeal function maps agents valuations to
valuations that they might instead have chosen
to report by lying. This function is
Computationally bounded.
68
Constrained Mechanisms

• The designer is not always free to design any
  mechanism.
• Social rules/laws can restrict the mechanism
  design.

69
Constrained Mechanisms

• Assumed that once a social law is imposed, the
  agents follow it.
• Now relax this assumption.
• Allow agents to enter into contracts amongst
  themselves.
• Assumes that the center can impose arbitrary
  fines on law breakers.
• Allow the center to bribe agents to act in a
  certain way.
• How can the center bias the outcome toward the
  desired outcome while minimizing costs.
• Allow the center to act on behalf of the agents.

70
Constrained Mechanisms

• Contracts
• During a game G the center
• Proposes a contract
• Gathers signatures
• Monitors the agents actions
• Fines any agent that deviates from the contract.
• How can the centers work be minimized?
• Center doesnt monitor the actions or participate
  in the contract signing.
• Contracts that are in equilibrium cause the
  center to sit idle.

71
Constrained Mechanisms

• Bribes
• Often requires a congestion game where the cost
  of a resource depends on how many other agents
  bid on that resource.
• The designer may want to prevent multiple agents
  from using the same resource.
• The mechanism can promise particular outcomes to
  the agents, often that match a dominant strategy.

72
Constrained Mechanisms

• Mediators
• An active center that acts on behalf of all
  agents.
• If all agents use the moderator, then there is a
  certain payoff for each agent. If only some
  agents use the moderator, then the moderator
  plays the favorable action.
• Can result in a strong equilibrium for the agents
  that use the moderator and guarantee that no
  coalition deviate.
• But, strong equilibrium are rare!

Even more rare when introduce moderators.