Communication and Prices in Economic Mechanisms

1
Communication and Prices in Economic Mechanisms

• Ilya Segal

2
Four lectures

1. Communication costs in economics
2. Communication and prices in social choice
3. Market design applications
4. Communication cost of incentives

3
Communication costs in economics
4
Market Design Problems

• Find allocations satisfying social goals when
  agents have private knowledge (e.g. of own
  preferences)
• Combinatorial auctions FCC spectrum, landing
  slots,
• Matching candidates to jobs, internships,
• Kidney exchange
• Incentives?
• can be verified with a Direct Revelation
  Mechanism (e.g., Vickrey-Clarke-Groves)
• But full revelation is costly and is avoided in
  most real-life mechanisms
• Instead sequential mechanisms e.g., iterative
  auctions, interviews, - to grope for equilibrium

5
Hayeks (1945) Critique of Lange-Lerner socialism

• Knowledge of particular circumstances of time
  and place too enormous to transmit to a central
  planner
• Ultimate decisions must be left to the people
  familiar with these circumstances
• Information needed to coordinate individual
  actions can be summarized in prices
• Nobody has yet succeeded in designing an
  alternative system

6
Costs of Revelation

• Communication costs
• E.g., combinatorial auctions 2L bundle values
• Measured in bits (communication complexity) or
  real numbers (message space dimension)
• Cost of agents evaluating own preferences
• E.g., job interviews, flyouts,
• More deviations when commitment is imperfect
  (short-term contracts, cheap talk)
• E.g., revealed info may be exploited by
  auctioneer or competitors in the future

7
Communication

• N set of agents, X set of outcomes
• Agent i privately observes his type Ri 2 Ri
• State R (R1, , RN) 2 R1 RN R
• Want to implement choice rule F R ? X
• F(R) optimal outcomes in state R
• Sequential communication may save over
  simultaneous
• Communication Protocol
• 1. Extensive-form message game
• 2. Outcomes assigned to terminal nodes
• 3. Agents strategies (type-contingent)
• Do not require incentive-compatibility (e.g.,
  agents could be computers)

8
Communication
Agent 2s type
Agent 1s type

• State space partitioned into product sets
• In every state R must implement an outcome in
  F(R)
• Characterizing such protocols is hard

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Verification (Nondeterminism)

• Omniscient oracle knows state R but must prove to
  an outsider that x 2 F(R)

• Announces message m 2 M
• Each agent accepts or rejects m based on his own
  type
• Acceptance by all agents verifies x 2 F(R)
• Verification in each state R

Agent 2s type
x
Agent 1s type

10
Verification Formalism

• Protocol ? ?M, µ, h ?
• M is a message space
• µ R ? M is a message correspondence s.t.
  Privacy Preservation
• µ(R) \i µi(Ri) ?R
• h M ? X is outcome function
• ? verifies F if ? ? h(µ(R)) ½ F(R) ?R
• ? fully verifies F if h(µ(R)) F(R) ?R

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Why Verification?

• Any deterministic protocol can be verified by
  oracle sending messages in agents stead (i.e., M
  terminal nodes of the protocol)
• ? verification costs communication costs
• Economic example Walrasian equilibrium message
  (allocation, prices)
• Steady state of a communication process (e.g.,
  tatonnement, auctions, deferred acceptance
  algorithms)?

12
Measuring communication costs

• Discrete number of bits (communication
  complexity)
• E.g., oracle needs log2M bits to encode message
• Continuous number of real numbers
• How many reals are needed to encode a message
  from M dimension of M?
• Agents costs of evaluating own types
• E.g., number of job interviews, dates, sorting
• Privacy reducing revelation of agents types
  E.g., revealed info may be exploited in the future

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Measuring Dimension Problem

• Intuition dimension of M number of real
  numbers needed to describe points in M

• Problem - Peano space-filling curve 0,1 ?
  0,1n
• Inverse Peano function describes 0,1n with one
  number
• Note Peano function is continuous (its inverse
  is not)

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Questions

• How to define the dimension of a space?
• Which transformations preserve this dimension?
• Two classes of approaches
• Topological dimensions
• Metric dimensions

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Topological dimension - inductive

• We say that space is 3-dimensional because the
  walls of a prison are 2-dimensional
• M is topological space (defined open sets)
• ind(?) - 1
• ind M ? k if every point in M lies in an open
  set whose boundary has ind ? k 1
• ind M min. integer k s.t.ind M ? k

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Preservation of topological dimension

• Proposition If f X ? Y is a continuous 1-to-1
  function then ind Y ? ind X
• Proof idea f(X) has fewer open sets than X
  harder to imprison points
• Could have ind Y gt ind X if f -1 is not
  continuous (e.g., Peano function)

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Metric dimension (ball covering)

• M is a metric space (has distances)
• NM(?) min. number of ?-balls needed to cover M
• E.g. N 0,1n (?) A ? (1/?)n
• In general, dim M limsup??0 ln NM(?)/ln (1/?)
• To cover unbounded S could allow to vary ball
  size (Hausdorff dimension)
• dim M ? ind M
• M irrational numbers in 0,1 ? dim M 1, ind
  M 0
• dim M may be fractal
• e.g., dim (Chinas coastline) 1.16

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Interpretation of metric dimension Approximation
with bits

• NM (?) min. number of messages needed to
  approximate points from M within ? (discretize)
• log2NM (?) min. number of bits needed to encode
  such messages
• Thus can approximate points from M within ? using
  dim M ? log2(1/?) bits as ??0
• Thus dim M relates to communication complexity of
  approximating M with bits
• But dim M may not represent the hardness of
  approximation for a fixed ? gt 0, or for ??0
  slowly as other parameters grow (will have
  example)

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Preservation of metric dimension

• Proposition If f X ? M is a 1-to-1 function and
  f -1 is Lipschitz
• (i.e., ?A gt0 dist(x, x?) ? A?dist(f (x), f
  (x?)) ?x,x?)
• then dim M ? dim X
• Proof f does not shrink distances except by a
  factor
• Continuity of f -1 not enough (e.g. let f be
  inverse Peano)

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Fooling set dimension bounds

• Rf ? R is a fooling set if ?R,R? ? Rf, R?R? ?
  ?(R)? ?(R?) ?
• Proposition Suppose Rf is a fooling set.
• (a) If ? has a (local) continuous selection, ind
  M ? ind Rf.
• (b) If ? has a (local) selection whose inverse is
  Lipschitz continuous, then dim M ? dim Rf.

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So, which dimension is better?

• Use topological dimension, restrict ? to have a
  continuous selection?
• Mount-Reiter, Walker, etc.
• Rules out discrete messages (e.g., indivisible
  allocations)
• Use metric dimension, restrict ? to have a
  selection whose inverse is Lipschitz continuous?
• Hurwicz 1977, Nisan-Segal
• Interpretation small errors in message
  transmission should not yield large mistakes
• Need not rule out any mechanisms - can define a
  right metric on any M to have Lipschitz
  continuity
• Allows fast discrete approximations

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Lecture 2 Communication and Prices in Social
Choice
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Hayeks (1945) Critique of socialism

• Knowledge of particular circumstances of time
  and place too enormous to transmit to a central
  planner
• Ultimate decisions must be left to the people
  familiar with these circumstances
• Information needed to coordinate individual
  actions can be summarized in prices
• Nobody has yet succeeded in designing an
  alternative system

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Is it necessary to use prices?

• Fundamental Welfare Thms Supporting prices (1)
  are sufficient for efficiency and (2) can be
  constructed with full information about the
  economy
• But then can compute efficient allocation
  directly
• Hurwicz, Mount-Reiter In a convex economy with
  distributed preference information, Walrasian
  equilibrium is dimensionally minimal among
  regular mechanisms verifying efficiency
• Did not rule out mechanisms that dont use prices
• Inapplicable to market design
• Nonconvexities, discrete decisions
• Other goals approximation, group stability,
  fairness,
• Other communication costs bits, evaluations,

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Informativeness Partial Order

• Message m is less informative than message m' if
  m' accepted ? m accepted.
• m is a minimally informative message verifying
  outcome x if any less informative message
  verifying x is equivalent to m.
• Such messages minimize communication costs
• Size of M - number of bits or reals to encode a
  message
• Preference evaluation costs, privacy loss, etc.

x
m'
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Allocate an Object between 2 Agents

• Messages verifying 2
• Minimally informative messages verifying 2

2
m
1

• Equivalent to announcing supporting equilibrium
  price p
• Each p must be used (in a diagonal state) ? need
  infinitely many messages (continuum)

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General Results

• Characterize social choice problems (social goals
  and preference domains) for which it is necessary
  to find supporting prices
• Algorithm deriving the form of prices (budget
  sets) that verify solutions to a given problem
  with minimal information
• Price space yields communication cost
• Selected applications
• Pareto efficiency in convex economies
• Exact or approximate surplus-maximization
• Stable many-to-one matchings
• Extra communication cost of incentives

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Social Choice Problem

• N set of agents, X set of outcomes
• Agent is type is a preference relation Ri over
  X
• State R (R1, , RN) 2 R1 RN R
• Choice rule F R ? X
• F(R) optimal outcomes in state R
• Protocol verifies F if ?R 2 R ?x 2 F(R)
  ?m 2 M that is acceptable in state R and
  verifies x

?
fully
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Verification by budget equilibria
x

• Oracles message
• Proposed outcome x ? X
• Budget set Bi ? X for each agent i
• Agent i accepts iff x is his optimal choice from
  Bi
• i.e., Bi ? L(x, Ri) ?y 2 X x Ri y
• Acceptance by all agents (equilibrium) must
  verify x

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Verification by budget equilibria
x

• Full verification ?R 2 R ?x 2 F(R)
• ? budget equilibrium (B,x) verifying x in state
  R
• E.g., Fundamental Welfare Theorems Any Pareto
  efficient allocation in a convex economy can be
  verified with a Walrasian equilibrium
• Extend to other social choice problems?

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Verification by budget equilibria
x

• Larger budget sets ? more informative equilibrium
• Definition F is monotonic if ?R?R ?x?F(R)
• ?R'?R, L(x, Ri) ? L(x, Ri') ?i ? x?F(R').
• Theorem F is fully verified with a budget
  equilibrium protocol ? F is monotonic.
• Williams (1986), Greenberg (1990), Miyagawa
  (2002)

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Necessity of budget equilibria?
x

• Knowing R, could compute an optimal outcome
  directly, without using supporting budget sets
• Can we construct budget equilibrium not just with
  full info, but from any verifying communication?

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Necessity of budget equilibria?
x
ÅR2m L( x, R2)
ÅR2m L( x, R1)
Definition F has the Budget Revelation Property
if for any message verifying x there exists a
less informative budget equilibrium (B,x)
verifying x. Theorem F satisfies BRP ? F is
Intersection-Monotonic (stronger than
monotonicity).
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Some choice rules satisfying BRP

• Pareto efficiency
• Approximate Pareto efficiency
• The core
• Stable matching
• Envy-free
• More generally Any CU rule described by
  coalitions blocking sets ?(x,S) ? X outcomes
  coalition S ? N can use to block candidate
  outcome x 2 X
• x ? F(R) ? no coalition S ? N has a strict Pareto
  improvement over x within ?(x,S)

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Venn Diagram
M verified with budget protocol
Pareto  
IM BRP
CU
Approx. Pareto  
Core  
No-Envy  
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Social Choice Rules Boolean Representation

• x ? F(R) can be written as a boolean formula
  with R atoms z Ri yy,z ? X, i ? N describing R
• Using conjunctions (? AND), disjunctions (?
  OR), negations (? NOT)
• F is monotonic ? need only use atoms
  x Ri yy ? X, i ? N , no negations
• ? can be written in Monotone Disjunctive Normal
  Form ?? ? ? (? (i,y)?? (x Ri y))
• ? ?R? F-1(x), MDNF can be verified with one of
  its clauses ? - a budget equilibrium (cf.
  Theorem 1)

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Restrictions imposed by IM

• Equivalently, for monotonic rules, x ? F(R) can
  also be written in Monotone Conjunctive Normal
  Form ?? ? ? (?(i,y)?? (x Ri y))
• F is Intersection Monotonic ? can be written as
  MCNF whose disjunctive clauses ? dont contain
  (x Ri y) ? (x Ri z) with y ? z
• Intuition if x ? F(R) is preserved by taking y
  or z individually out of L(x,Ri), should be
  preserved by taking out y and z together

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Restrictions imposed by CU

• F if CU ? x ? F(R) can be written as MCNF
  ?? ? ? (?(i,y)?? (x Ri y)) s.t. each clause ? has
  a single y
• i.e., ? takes the form ?i?S (x Ri y)
• Interpretation coalition S does not want to
  block x with y
• i.e., does not contain (x Ri y)?(x Rj z) with y ?
  z
• Characterizes monotonic choice rule that are
  binary, which means
• calculate social relation S between ?x,y from
  agents preferences x Ri y (e.g., S not
  blocked by) IIA
• Choose maximal elements in S from X
• Note if we required S to be rational we would
  hit Arrows Impossibility Theorem

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Lecture 3 Market Design Applications
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Necessity of budget equilibria
x

• Definition F has the Budget Revelation Property
  if for any message verifying x there exists a
  less informative budget equilibrium (B,x)
  verifying x.
• Theorem F satisfies BRP ? F is
  Intersection-Monotonic.
• Stronger than monotonicity, but satisfied by many
  choice rules e.g. Pareto, approx. Pareto, core,
  stability, envy-free all CU rules.

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Intuition

• Social goals congruent with private preferences
  ? minimize communication by asking agents to act
  on their knowledge selfishly within budget sets
  (cf. Hayek)
• Design budget sets to coordinate choices and
  attain social goals with minimal communication

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Minimally Informative Verifying Messages
R
x

• Budget Revelation Property ? use budget
  equilibria
• Minimize informativeness shrink budget sets
• ? critical states R
• Bi L(x, Ri) \ R'i ?Ri x?F(R'i, R-i) L(x,
  R'i) ?i
• In such R, (B1,, BN, x) is a unique budget
  equilibrium verifying x (up to equivalence)

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Market Design Roadmap

• Use budget-shrinking algorithm to construct min.
  informative budget equilibria (B,x) verifying x
• If Bi L(x, Ri) ?i for some R 2 R (critical
  state) ? minimal message space for full
  verification of F
• To bound below simple verification cost - use
  tricks, e.g.,
• Restrict to states R in which F(R) is a singleton
• Construct a fooling set whose states cant share
  a verifying budget equilibrium
• Verification cost very high ? problem is hopeless
• Verification cost low ? Can it be achieved in a
  deterministic, incentive-compatible protocol?

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Application Pareto Efficiency in Smooth Convex
Exchange Economies

• L goods Ri over xi 2 RL convex, smooth, with a
  positive utility gradient

45

• Proposition. m is a minimally informative message
  verifying the Pareto efficiency of an interior
  allocation in a smooth convex economy ? m is
  equivalent to a Walrasian equilibrium.
• Any such equilibrium (B, x) is a unique Walrasian
  equilibrium supporting x
• ? Number of variables for full verification
  (N-1)L quantities (L 1) prices

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Verification Cost

• Fix endowments ? (N-1)(L-1) quantities (L 1)
  prices N(L-1) numbers
• Can we verify Pareto efficiency with fewer
  numbers?
• Fooling set of Cobb-Douglas economies (Hurwicz)
• Utilities ui (xi) ?l xilail with Sl ail 1
  for all i
• Described with N(L -1) parameters
• FOC for (p, x) being a Walrasian eqm
• ail/aik (pl xil ) /(pk xik) for all l,k,I
• No two distinct Cobb-Douglas economies share a
  Walrasian eqm ? give a fooling set for Pareto
  efficiency
• What about deterministic communication?
• Tatonnement converges fast in some but not all
  economies

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Pareto Efficiency with Numeraire

• x (k, t1,,tN ) feasibility Si ti 0
• Ri 2 Ri given by quasilinear utility ui(k)ti
• Efficiency maxk2K Si ui(k) (total surplus)

t2
t1
k
48

• Proposition. m is a minimally informative message
  verifying efficiency with numeraire ? m is
  equivalent to a price equilibrium with
  personalized nonlinear prices p 2 ? NK s.t.
  ?i pi (k) const on k 2 K.
• Each such price must be used in the critical
  state where Si ui(k) const on k 2 K
• ? Cost (N 1)(K 1) real numbers full
  revelation of N 1 agents valuations
• Combinatorial auctions N 2, K 2L ?
  communication cost 2L 1 numbers

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Stable Many-to-one Matching

• N consists of firms (F) and workers (W)
• Matching x binary relation from F to W in which
  a worker matches with at most one firm
• Ri 2 Ri depends only on is partners

x
50
Stable Matching

• Matching is stable if these coalitional
  deviations are not strictly Pareto improving
• Firm hires some new workers (and fires some)
• Worker quits to become unemployed
• This describes an IM choice rule

x
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Minimally Informative Equilibria

• A workers budget set described by available
  firms
• A firms budget set available groups of workers
• In a minimally informative equilibrium verifying
  stability, the groups consist of
• The firms current workers
• Workers who dont have the firm in their budget
  sets
• ? firms budget sets, non-combinatorial

x
52

• Lemma. m is a minimally informative message
  verifying the stability of matching x ? m is
  equivalent to a partitional equilibrium, in which
  each off-equilibrium match is in either partners
  budget set, but not both.
• Any such equilibrium (B, x) is a unique
  partitional equilibrium supporting x in state R
  in which Bi L(x, Ri) ?i.
• Cost of full verification 2FW bits
• Cf. cost of describing a match x FW bits
• Cf. describing firms preferences over 2W groups
  log2(2W)! W ?2W bits
• Two questions
• Is the cost of verification any lower?
• Is the deterministic communication cost higher?

53
Verification cost is the same

• Take any partitional equilibrium (B, x), and
  critical state R s.t. Bi L(x, Ri) ?i
• Further restrict R to ensure that x is unique
  stable match
• Each i strictly prefers his current match within
  Bi
• Each i strictly prefers quitting to getting a new
  partner from Bi
• Cant have a stable match x? ? x
• i gets a new partner from Bi ? i would quit
• i gets a new partner j ? Bi ? i 2 Bj ? j would
  quit
• x? ? x ? x? is in all budget sets ? a coalition
  would block x? by x
• ? (B, x) must be used for verification in state R
• ? Communication cost 2FW bits
• Same logic works for 1-to-1 matching (cost FW
  bits)
• This bounds below deterministic communication
  cost

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Deterministic Communication

• This lower bound is almost achieved for
  substitutable preferences by Gale-Shapley
  deferred acceptance algorithm
• ( 2FW steps matches offered, rejected)
• Exponentially less than full revelation of firms
  combinatorial preferences
• The algorithm also minimizes evaluation costs of
  the responding side (whose budget sets are
  minimized)
• With uniformly drawn preferences, on average only
  1/3 of potential partners evaluated

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Determinism vs. Nondeterminism

• Recall Deterministic CC in bits
  nondeterministic CC in bits
• Gap is at most exponential e.g., try each of
  the M oracles messages for acceptance instead of
  encoding with log2M bits
• Gap is small in some cases e.g., matching or
  auctions with substitute preferences - monotonic
  tatonnement converges quickly
• But there are IM rules with exponential gap

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Example of Exponential Gap

• 2 agents hire 2 out of 3m workers
• Agent 1 is happy if the hired workers share a
  language
• Agent 1 knows privately each workers language
• Public knowledge all workers are monolingual, m
  languages spoken by a pair of workers, and m
  languages spoken by a single worker.
• Agent 2 knows privately 2m1 capable workers
• Agent 2 is happy 1 if both hired workers are
  capable
• Social goal make both agents happy. - CU, IM.
• Such pair always exists and can be verified by
  its announcement - 2 log2 (3m) bits
• But deterministic CC is asymptotically
  proportional to m
• Equivalent to the Pair-Disjointness.problem

57
Average-Case Goals

• E.g., approximate expected surplus, given a
  probability distribution over states
• Example discrete public good, N agents i.i.d.
  values vi 20,1, Prvi 1 ?, Cost c lt ?N.
• N large ? building is efficient with prob.? 1
• Verifying efficiency requires finding c agents to
  charge Lindahl price 1, takes c logN bits
  
• for ? -approximation guarantee, (k-?) log2N bits
• Government solution approximates expected
  efficiency without any communication, prices
  (cf. Groves-Hart)
• Another Example with many decisions, authority
  may approximate expected efficiency when finding
  prices would take exponentially longer (cf.
  Coase, Simon)
• Examples known in which expected-surplus
  approximation is hard to attain e.g.,
  combinatorial auctions

58
Economics without Incentives?

• Thought experiment people are honest. Would
  basic economics institutions (markets, firms)
  remain?
• Minimize communication by asking people to pursue
  self-interest guided by prices
• ? Scope of market design
• ? Required price space
• Lower bounds on communication and evaluation
  costs
• Average-case goals may yield non-market
  institutions (governments, firms)

59
Lecture 4Communication Cost of Incentives
60
So far

• Can calculate communication costs of many
  economic problems
• E.g., use the Budget Revelation Property,
  construct min. informative verifying budget
  equilibria
• In some cases, cost is prohibitive, ? cost of
  full revelation of preferences
• E.g., combinatorial auctions with general
  valuations need whole combinatorial price space
  
• In other cases, cost is manageable
• E.g., auctions or matching with substitute
  preferences enough to use individual
  prices/budget sets
• Can we then construct an incentive-compatible
  mechanism?

61
Incentives and Prices

• Budget set in mechanism design set of agents
  attainable outcomes
• E.g. Menus Taxation Principle
• Incentives agent maximizes utility within his
  budget set
• Remarkable even with no incentives (truthful
  agents), must still describe budget sets, ask
  agents to maximize own utility within them
• For social goals that are congruent with
  preferences

62
Nash implementation

• Agents preferences commonly known (but not to
  designer)
• Mechanism g M1? ? MN ? X
• (Full) implementation Set of Nash Equilibria
  F(R) ?R 2 R
• m is NE of g ? (B, x) is a budget equilibrium,
  where xg(m), Bi g(mi?,m-i) mi? 2 Mi ?i
• ? A Nash protocol is a budget protocol
• Theorem 1 ? Any Nash implementable F is monotonic
  
• Even with symmetric info, selfishness ? reveal
  budget sets
• Proposition Communication cost of Nash
  implementation ? communication cost of full
  realization
• May not hold for extensive-form mechanisms (in
  which strategies m are not revealed)

63
Communication Cost of (1-stage) Nash
implementation

• Monotonicity ? Nash implementability
  (not every budget equilibrium protocol is a Nash
  protocol)
• But is almost sufficient
• Definition F has No Veto Power (NVP) if i
  xRiy ?y 2 Y ? N 1 ? x 2 F(R).
• Trivial in economic applications with N ? 3
• Maskin (1977) Any monotonic NVP choice rule is
  Nash implementable
• Proposition If F is IM and NVP,
  communication cost of Nash implementation ? N
  ? (communication cost of full realization)
  log2N.

64
Proof Mechanism

• Let ? minimal space of budget equilibria fully
  verifying F
• Each agent i announces (Ei, li), where Ei
  (B1i,, BNi, xi) 2 ?, li 2 1,, N
• E1 EN (B1,, BN, x) ? implement x
• Ej (B1,, BN, x) ?j?i and xi ? Bi ? implement x
• Ej (B1,, BN, x) ?j?i and xi 2 Bi ? implement
  xi
• Otherwise implement xi for i (?j lj ) modN 1
• x 2 F(R) ? all agents announcing the same
  equilibrium (B1,, BN, x) 2 ? in state R is a NE
• Case-1 NE (B1,, BN, x) verifies x, is a budget
  eqm in state R ? x 2 F(R)
• Other cases all but (perhaps) one agent could
  deviate to Case 4 to get his best outcome ? by
  NVP, x 2 F(R)

65

• May further reduce cost e.g. by letting Bi be
  announced only by agent i modN1
• McKelvey, Reiter-Reichelstein
• Key each agents budget set must be determined
  by others messages prevent price
  manipulation
• With private information, incentives require
  constructing Bi without using is info ? extra
  communication cost

66
Communication Cost of Selfishness (with Ron Fadel)

• Incentive-Compatibility with private information
• Ex Post IC Each agents strategy is optimal
  given others even if he knew other agents types
• Weaker than Dominant-Strategy IC in an extensive
  form, dont consider others unused strategies
• Bayesian IC Each agents strategy is optimal
  given others on expectation over their types

67
Communication Cost of Selfishness

• CCS (Minimal Communication Cost of an
  Incentive-Compatible mechanism) - Communication
  Complexity
• Related work Reichelstein 1984, Green and
  Laffont 1987, Lahaie and Parkes 2004, Feigenbaum
  el al. 2004, Johari 2004

68
Setup

• Quasilinear payoffs ui(y)ti, with y 2 Y
• ui 2 Ui ½ ?Y - private type of agent i
• Types drawn independently (basic model)
• Decision function f U1 UI ! Y
• Dont care about transfers

69
Communication Protocols
y1
y2
u1'
u1'
u1
u1
y3
y4
y5
Communication cost
3 bits (worst-case)

• Binary extensive-form message game (tree)
• Agents (type-contingent) strategies
• Outcomes assigned to leaves

70
Incentivizing a Protocol
, t11, t21
y1
, t12 , t22
y2
2
1
, t13, t23
1
y3
, t14, t24
y4
2
, t15, t25
y5

• Assign incentivizing payments to the leaves
• Hide history (create Information Sets) to prevent
  contingent deviations
• A given protocol may not be incentivizable may
  need more communication

71
Sources of CCS

• EPIC No need to hide info CCS comes from the
  need to compute payments
• BIC CCS comes from need to hide info (computing
  payments does not require extra bits)
• Restrict to f that is implementable in some IC
  mechanism (e.g. full revelation)
• for others, CCS 1

72
A Protocol that cant be EPIC incentivized

• Allocate one object between two agents
• efficiently values v1 2 1,2,3,4, v2 2 0,5.
• Protocol
• Agent 1 announces v1 (2 bits)
• Agent 2 takes iff v2 v1 (1 bit)
• Intuition for EPIC, agent 1 must be charged a
  price within 1 of v2, but it is not revealed
• Can show that any EPIC protocol must take 3
  bits ? EPIC CCS 1
• A similar example for BIC

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An Upper Bound on CCS(both BIC and EPIC)

• Take a protocol P computing f (which is
  implementable)
• Can incentivize agents if observe their
  strategies in P
• Transfers need only depend on the strategies
• Punish strategies that are not consistent with
  any type
• Protocol in which all agents simultaneously
  announce their strategies in P is incentivizable

74
An Upper Bound on CCS(both BIC and EPIC)
y1
y2
2
1
1
y3
y4
2
y5

• A d-bit protocol P has 2d decision nodes ?
  agents strategies can be described with 2d
  bits
• ? Comm. Complexity with IC 2Comm. Complexity
• Is the bound tight? For EPIC, open question

75
Example Low CCS for Efficiency

• f(u) solves maxy ?i ui(y)
• Let agents announce final utilities wi ui(y),
  pay each agent ti ?j?i wj
• ? Agents become a team
• ? Efficient protocol is EPIC
• ? Even if cant reach full efficiency, strategies
  maximizing expected surplus form a BNE
• Also, any EPIC implementable f (e.g., efficiency)
  has BIC CCS 0

76
Exponential Bound reached for BIC
Boss

• Expert privately knows 1-to-1 mapping between K
  decisions and K consequences
• Also has a private utility over decisions
• Boss privately knows desired consequence

77
Bound is reached for BIC
Expert
4
5
2
3
1
Decisions
Consequences
Boss

• A simple protocol
• Boss announces desired consequence (logK bits),
• Expert decides (logK bits)
• Not BIC Expert will maximize own utility

78
Bound is reached for BIC
Boss

• A BIC protocol
• Expert reveals mapping
• ( log(K!)KlogK bits exponentially longer)
• Boss decides
• We show that any BIC protocol takes K/2 bits
  - exponentially longer than the simple protocol

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Extensions CCS is unbounded for

• Average-case communication complexity
• Example allocate object efficiently between 2
  agents with values uniformly drawn from 0,1
• Bisection takes on expectation 4 bits (Arrow et
  al.)
• EPIC for Agent 1 requires essentially charging
  him agent 2s value, which has unbounded entropy
• BIC CCS with Correlated Types
• Example Agent 1 has large type that determines
  binary outcome and correlates with Agent 2s type
  
• BIC must punish Agent 1 when caught lying by
  Agent 2
• EPIC CCS with Interdependent Valuations
  vi(x,si,s-i), where si is agent is type
• Example Agent 1 knows whether he should get the
  object but his value for it is known only to
  Agent 2
• EPIC requires charging Agent 1 this value

80
Open Questions

• EPIC CCS how high can it be?
• In what practical problems is CCS low?
• Can CCS be reduced substantially if ICs only need
  to be satisfied approximately (equivalently,
  utilities given with finite precision)?

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Ideas for understanding firms

• Coase (1937), Simon (1951) firms are islands of
  authority where discovering what the relevant
  prices are is too costly
• Have example where authority achieves efficiency
  with probability ? 1, but verifying it (finding
  prices) takes exponentially more bits
• Communication may be distributed to economize on
  individual costs
• Individual costs may be reduced by hiring extra
  agents (managers) to relay prices (cf.
  computational models of Radner - van Zandt)