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Market Communication Christopher Wilkens in Communication - - PowerPoint PPT Presentation

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Market Com- munication in Production Economies Market Communication Christopher Wilkens in Communication Complexity Production Economies Market Communication Markets Arrow-Debreu Markets Christopher Wilkens Related Work DPS 2002

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SLIDE 1

Market Com- munication in Production Economies Christopher Wilkens Communication Complexity

Market Communication

Markets

Arrow-Debreu Markets Related Work DPS 2002 Large Endowments Main Result

Conclusion

Market Communication in Production Economies

Christopher Wilkens

UC Berkeley

WINE ’10

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Market Com- munication in Production Economies Christopher Wilkens Communication Complexity

Market Communication

Markets

Arrow-Debreu Markets Related Work DPS 2002 Large Endowments Main Result

Conclusion

A Communication Bound for Market Equilibrium

In a market with m goods, m prices are sufficient to communicate an equilibrium...

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SLIDE 3

Market Com- munication in Production Economies Christopher Wilkens Communication Complexity

Market Communication

Markets

Arrow-Debreu Markets Related Work DPS 2002 Large Endowments Main Result

Conclusion

A Communication Bound for Market Equilibrium

In a market with m goods, m prices are sufficient to communicate an equilibrium... BUT we show the number of bits of information must be polynomial in

m, n, the number of agents, and l, the number of production firms.

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Market Com- munication in Production Economies Christopher Wilkens Communication Complexity

Market Communication

Markets

Arrow-Debreu Markets Related Work DPS 2002 Large Endowments Main Result

Conclusion

Communication Complexity

Yao’s 2-party model:

Alice has a private, n-bit string a Bob has a private, n-bit string b Alice and Bob want to compute f (a, b) How many bits of communication are required to compute f (a, b)?

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Market Com- munication in Production Economies Christopher Wilkens Communication Complexity

Market Communication

Markets

Arrow-Debreu Markets Related Work DPS 2002 Large Endowments Main Result

Conclusion

Communication Complexity

Yao’s 2-party model:

Alice has a private, n-bit string a Bob has a private, n-bit string b Alice and Bob want to compute f (a, b) How many bits of communication are required to compute f (a, b)?

Example: Alice and Bob have strings a and b. Are a and b different?

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Market Com- munication in Production Economies Christopher Wilkens Communication Complexity

Market Communication

Markets

Arrow-Debreu Markets Related Work DPS 2002 Large Endowments Main Result

Conclusion

The Market Equilibrium Story

...but market equilibrium doesn’t fit this pattern:

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Market Com- munication in Production Economies Christopher Wilkens Communication Complexity

Market Communication

Markets

Arrow-Debreu Markets Related Work DPS 2002 Large Endowments Main Result

Conclusion

The Market Equilibrium Story

...but market equilibrium doesn’t fit this pattern:

1 The “Invisible hand” publishes prices.

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Market Com- munication in Production Economies Christopher Wilkens Communication Complexity

Market Communication

Markets

Arrow-Debreu Markets Related Work DPS 2002 Large Endowments Main Result

Conclusion

The Market Equilibrium Story

...but market equilibrium doesn’t fit this pattern:

1 The “Invisible hand” publishes prices. 2 Agents sell endowment to the market.

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SLIDE 9

Market Com- munication in Production Economies Christopher Wilkens Communication Complexity

Market Communication

Markets

Arrow-Debreu Markets Related Work DPS 2002 Large Endowments Main Result

Conclusion

The Market Equilibrium Story

...but market equilibrium doesn’t fit this pattern:

1 The “Invisible hand” publishes prices. 2 Agents sell endowment to the market. 3 Firms maximize profit.

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SLIDE 10

Market Com- munication in Production Economies Christopher Wilkens Communication Complexity

Market Communication

Markets

Arrow-Debreu Markets Related Work DPS 2002 Large Endowments Main Result

Conclusion

The Market Equilibrium Story

...but market equilibrium doesn’t fit this pattern:

1 The “Invisible hand” publishes prices. 2 Agents sell endowment to the market. 3 Firms maximize profit. 4 Agents buy optimal bundle given market prices.

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SLIDE 11

Market Com- munication in Production Economies Christopher Wilkens Communication Complexity

Market Communication

Markets

Arrow-Debreu Markets Related Work DPS 2002 Large Endowments Main Result

Conclusion

The Market Equilibrium Story

...but market equilibrium doesn’t fit this pattern:

1 The “Invisible hand” publishes prices. 2 Agents sell endowment to the market. 3 Firms maximize profit. 4 Agents buy optimal bundle given market prices. 5 The market clears.

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Market Com- munication in Production Economies Christopher Wilkens Communication Complexity

Market Communication

Markets

Arrow-Debreu Markets Related Work DPS 2002 Large Endowments Main Result

Conclusion

Market Communication

Deng, Papadimitriou, and Safra (STOC 2002) Setting: n agents, each agent i ∈ [n]...

has private information xi ∈ Xi, and must compute fi(x1, . . . xn).

Agent 0, the“invisible hand,” knows (x1, . . . xn).

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Market Com- munication in Production Economies Christopher Wilkens Communication Complexity

Market Communication

Markets

Arrow-Debreu Markets Related Work DPS 2002 Large Endowments Main Result

Conclusion

Market Communication

Deng, Papadimitriou, and Safra (STOC 2002) Setting: n agents, each agent i ∈ [n]...

has private information xi ∈ Xi, and must compute fi(x1, . . . xn).

Agent 0, the“invisible hand,” knows (x1, . . . xn). Protocol:

1 Agent 0 computes x0 = g0(x1, . . . xn) 2 Agent 0 broadcasts x0 3 Each agent i computes fi(x1, . . . , xn) = gi(x0, xi)

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Market Com- munication in Production Economies Christopher Wilkens Communication Complexity

Market Communication

Markets

Arrow-Debreu Markets Related Work DPS 2002 Large Endowments Main Result

Conclusion

Market Communication

Definition A market communication protocol is a set of functions (g0(·), g1(·), . . . gn(·)) where g0 : X1 × . . . Xn → X0, and gi(g0(x1, . . . xn), xi) = fi(x1, . . . xn). The amount of market communication is the size of x0 = g0(x1, . . . xn).

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Market Com- munication in Production Economies Christopher Wilkens Communication Complexity

Market Communication

Markets

Arrow-Debreu Markets Related Work DPS 2002 Large Endowments Main Result

Conclusion

An Arrow-Debreu Market

m tradable, divisible goods n agents with...

utility functions ui : Rm

+ → R

endowment ei ∈ Rm

+

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Market Com- munication in Production Economies Christopher Wilkens Communication Complexity

Market Communication

Markets

Arrow-Debreu Markets Related Work DPS 2002 Large Endowments Main Result

Conclusion

An Arrow-Debreu Market

m tradable, divisible goods n agents with...

utility functions ui : Rm

+ → R

endowment ei ∈ Rm

+

l production firms...

with a set of production possibilities Yk ⊂ Rm agent i owns a σi,k share of firm k

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Market Com- munication in Production Economies Christopher Wilkens Communication Complexity

Market Communication

Markets

Arrow-Debreu Markets Related Work DPS 2002 Large Endowments Main Result

Conclusion

Market Equilibrium

Equilibrium is tuple (x, y, π) such that

xi is an optimal consumption bundle: xi ∈ argmaxx|0≤π·x≤Mi ui(x) where Mi = π · ei +

k σi,k (π · yk)

yk is an optimal production plan: yk ∈ argmaxy∈Yk π · y The market clears: 0 ≤

  • i

xi ≤

  • i

ei +

  • k

yk

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Market Com- munication in Production Economies Christopher Wilkens Communication Complexity

Market Communication

Markets

Arrow-Debreu Markets Related Work DPS 2002 Large Endowments Main Result

Conclusion

Related Work

Dimensionality of message spaces

Economists’ analog to communication complexity m − 1 real numbers (prices) are optimal for representing a market equilibrium Calsamiglia (1988) — parametric communication (analogous to market communication) does not help.

Nisan and Segal (2006) — “prices” are required to communicate an allocation Deng, Papadimitriou, and Safra (2002) — Market communication

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Market Com- munication in Production Economies Christopher Wilkens Communication Complexity

Market Communication

Markets

Arrow-Debreu Markets Related Work DPS 2002 Large Endowments Main Result

Conclusion

DPS 2002

Theorem (Deng, Papadimitriou, and Safra, STOC 2002) If the non-satiation requirement for utilities is relaxed, communicating an exchange equilibrium in the market communication model requires Ω (n log(m + n)) bits of communication.

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Market Com- munication in Production Economies Christopher Wilkens Communication Complexity

Market Communication

Markets

Arrow-Debreu Markets Related Work DPS 2002 Large Endowments Main Result

Conclusion

DPS 2002

Theorem (Deng, Papadimitriou, and Safra, STOC 2002) If the non-satiation requirement for utilities is relaxed, communicating an exchange equilibrium in the market communication model requires Ω (n log(m + n)) bits of communication. Weaknesses: Relaxes non-satiation requirement. Proof uses Ω( n

m)-bit numbers in players’ endowments.

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Market Com- munication in Production Economies Christopher Wilkens Communication Complexity

Market Communication

Markets

Arrow-Debreu Markets Related Work DPS 2002 Large Endowments Main Result

Conclusion

Why Ω( n

m)-bit Endowments Matter

Lemma If eij is represented using β bits, then communicating a market equilibrium in the market communication model requires (m − 1)β bits of communication.

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Market Com- munication in Production Economies Christopher Wilkens Communication Complexity

Market Communication

Markets

Arrow-Debreu Markets Related Work DPS 2002 Large Endowments Main Result

Conclusion

Why Ω( n

m)-bit Endowments Matter

Lemma If eij is represented using β bits, then communicating a market equilibrium in the market communication model requires (m − 1)β bits of communication. β = Ω( n

m) implies Ω(n) bits of communication are necessary

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Market Com- munication in Production Economies Christopher Wilkens Communication Complexity

Market Communication

Markets

Arrow-Debreu Markets Related Work DPS 2002 Large Endowments Main Result

Conclusion

Why Ω( n

m)-bit Endowments Matter

Proof Sketch (m = n = 2). Agent 1...

Has 1 units of good A Only wants good B

Agent 2...

Has e2,B unit of good B Only wants good A

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Market Com- munication in Production Economies Christopher Wilkens Communication Complexity

Market Communication

Markets

Arrow-Debreu Markets Related Work DPS 2002 Large Endowments Main Result

Conclusion

Why Ω( n

m)-bit Endowments Matter

Proof Sketch (m = n = 2). Agent 1...

Has 1 units of good A Only wants good B

Agent 2...

Has e2,B unit of good B Only wants good A

In equilibrium, agent 1 gives all of A, gets all of B Agent 1 must learn e2,B, i.e. β bits of information.

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Market Com- munication in Production Economies Christopher Wilkens Communication Complexity

Market Communication

Markets

Arrow-Debreu Markets Related Work DPS 2002 Large Endowments Main Result

Conclusion

Main Result

Theorem In the market communication model of Deng, Papadimitriou, and Safra (STOC 2002), computing a market equilibrium requires 3 4m − 1

  • (β + lg(n − 1)) + n + l − O(1)

bits of communication. Moreover, if the number of possible utility functions and production sets is exponential in m, then 3 4m − 1

  • (β + lg(n − 1)) + (m − 2)(n + l) − O(m)

bits of information are required.

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Market Com- munication in Production Economies Christopher Wilkens Communication Complexity

Market Communication

Markets

Arrow-Debreu Markets Related Work DPS 2002 Large Endowments Main Result

Conclusion

Main Result: Strong Version

Lemma In the market communication model, for a market where ui ∈ U, Yk ∈ Y, and β is the number of bits used to represent each eij, computing a market equilibrium requires 3 4m − 1

  • (β + lg(n − 1))

+ (n − 1 − |U|

2 m−2 ) lg |U|

+ (l − 1 − |Y|

4 m ) lg |Y|

bits of communication. |U| = |Y| = 2 gives the first bound of the theorem, |U| = |Y| = 2m−2 gives the second bound.

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Market Com- munication in Production Economies Christopher Wilkens Communication Complexity

Market Communication

Markets

Arrow-Debreu Markets Related Work DPS 2002 Large Endowments Main Result

Conclusion

Main Result: Strengths

Gives a stronger bound than DPS02 without Ω(n)-bit endowments. Gives intuition and dependence on market parameters. Does not relax non-satiation assumption. Includes production.

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Market Com- munication in Production Economies Christopher Wilkens Communication Complexity

Market Communication

Markets

Arrow-Debreu Markets Related Work DPS 2002 Large Endowments Main Result

Conclusion

Main Result: Weaknesses

Works because real numbers rarely sum to the same value

Likely requires exact equilibria

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Market Com- munication in Production Economies Christopher Wilkens Communication Complexity

Market Communication

Markets

Arrow-Debreu Markets Related Work DPS 2002 Large Endowments Main Result

Conclusion

Proof Sketch: Utilities and Consumption

Proof Sketch. Agent i ≥ 2

has utility function ui(x) = xi,1 +

n

  • j=2

2

  • lg ci,j√xi,j

for ci,j ∈ C = {2, 3, . . . } (the first |C| primes) has an endowment ei where ei,j may be any β-bit number (j ≥ 2).

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Market Com- munication in Production Economies Christopher Wilkens Communication Complexity

Market Communication

Markets

Arrow-Debreu Markets Related Work DPS 2002 Large Endowments Main Result

Conclusion

Proof Sketch: Utilities and Consumption

Proof Sketch (Continued). Agent i = 1

has utility function ui(x) = xi,1 +

n

  • j=2

2√xi,j has a fixed (known) endowment ei.

Agent 1 has no private information.

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Market Com- munication in Production Economies Christopher Wilkens Communication Complexity

Market Communication

Markets

Arrow-Debreu Markets Related Work DPS 2002 Large Endowments Main Result

Conclusion

Proof Sketch: Utilities and Consumption

Proof Sketch (Continued). For j ≥ 2, equilibrium consumption by agent 1 will be x1,j =

  • i

ei,j 1

  • i lg ci,j

=

  • i ei,j

lg (

i ci,j)

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Market Com- munication in Production Economies Christopher Wilkens Communication Complexity

Market Communication

Markets

Arrow-Debreu Markets Related Work DPS 2002 Large Endowments Main Result

Conclusion

Proof Sketch: Utilities and Consumption

Proof Sketch (Continued). For j ≥ 2, equilibrium consumption by agent 1 will be x1,j =

  • i

ei,j 1

  • i lg ci,j

=

  • i ei,j

lg (

i ci,j)

  • i ci,j is prime factorization, so |C|n−1

|C|!

possibile values (n − 1)2β possible values for

i ei,j, so can show

(n − 1)2β × |C|n−1 |C|! possible values for x1,j.

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Market Com- munication in Production Economies Christopher Wilkens Communication Complexity

Market Communication

Markets

Arrow-Debreu Markets Related Work DPS 2002 Large Endowments Main Result

Conclusion

Proof Sketch: Utilities and Consumption

Proof Sketch (Continued). Aggregating over all j ≥ 2 and noting that |U| = |C|m−1, we have

  • (n − 1)2β × |C|n−1

|C|! m−1 ≥

  • (n − 1)2βm−1

|U|n−1−|U|

1 m−1

possible values for x1.

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Market Com- munication in Production Economies Christopher Wilkens Communication Complexity

Market Communication

Markets

Arrow-Debreu Markets Related Work DPS 2002 Large Endowments Main Result

Conclusion

Proof Sketch: Utilities and Consumption

Proof Sketch (Continued). Since agent 1 has no private information, there must be at least as many communication sequences as values of x1 Thus, (m − 1)(β + lg(n − 1)) +

  • n − 1 − |U|

1 m−1

  • lg |U|

bits of communication are required to reach equilibrium.

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Market Com- munication in Production Economies Christopher Wilkens Communication Complexity

Market Communication

Markets

Arrow-Debreu Markets Related Work DPS 2002 Large Endowments Main Result

Conclusion

Proof Sketch: Production

Proof Sketch (Continued). Firm k ≥ 2 has production function for goods j ≡ 1 mod 2: yj,k = fj,k(y1,k) = 2

  • y(j−1),k · lg cj,k

Agents only want goods j ≥ 2 Like consumption, m 2 (β + lg(n − 1)) +

  • n − 1 − |Y|

2 m

  • lg |Y|

bits of communication are required

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Market Com- munication in Production Economies Christopher Wilkens Communication Complexity

Market Communication

Markets

Arrow-Debreu Markets Related Work DPS 2002 Large Endowments Main Result

Conclusion

Proof Sketch: Finale

Proof Sketch (Continued). Putting the two results side-by-side in a single market requires m 2 − 1

  • (β + lg(n − 1))

+ (n − 1 − |U|

2 m−2 ) lg |U|

m 4 (β + lg(n − 1)) + (l − 1 − |Y|

4 m ) lg |Y|

bits of communication.

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Market Com- munication in Production Economies Christopher Wilkens Communication Complexity

Market Communication

Markets

Arrow-Debreu Markets Related Work DPS 2002 Large Endowments Main Result

Conclusion

Main Result, Reprieve

Theorem In the market communication model of Deng, Papadimitriou, and Safra (STOC 2002), computing a market equilibrium requires 3 4m − 1

  • (β + lg(n − 1)) + n + l − O(1)

bits of communication. Moreover, if the number of possible utility functions and production sets is exponential in m, then 3 4m − 1

  • (β + lg(n − 1)) + (m − 2)(n + l) − O(m)

bits of information are required.

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Market Com- munication in Production Economies Christopher Wilkens Communication Complexity

Market Communication

Markets

Arrow-Debreu Markets Related Work DPS 2002 Large Endowments Main Result

Conclusion

Conclusion

Lower bound tarnishes power of prices. A result for approximate equilibria could be more meaningful.

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Market Com- munication in Production Economies Christopher Wilkens Communication Complexity

Market Communication

Markets

Arrow-Debreu Markets Related Work DPS 2002 Large Endowments Main Result

Conclusion

Thank you.