Computation in Distributed Information Markets, Lecture Notes - Computer Science, Study notes of Computers and Information technologies

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Computation in a Distributed

Information Markets

Presented by Zhenming Liu

Information Aggregation Mechanisms

• Prediction Market - Example: Logarithmic scoring rule, betting on permutation, pricing tournament. - These examples are more relevant to the be- havior of the market.
• This Paper: study on a process whose goal is to jointly compute the value of a function. - A ”classy” model that is closely related to the filtration defined over a probability space. - Less focused on the behavior of the market, e.g., motivation to trade, the cost of the mar- ket.

• Model of an information market - There are n traders in the system; i th trader holds the bit xi. - All agents have a common prior on the distri- bution x = ( x 1 , ..., xn ). - Security being traded: a function f ( x ), where f is a common knowledge.
• More simplifying assumptions: - Each round every trader reports to the cen- tralized agent the E[ f ( x )| σ ], where σ is all the available information to that trader. - the centralized agent broadcasts the average value to all traders. - The market ”terminates” when there is no more trader can update E[ f ( x )| σ ]. - These assumptions may not be realistic.

• Model of an information market - There are n traders in the system; i th trader holds the bit xi. - All agents have a common prior on the distri- bution x = ( x 1 , ..., xn ). - Security being traded: a function f ( x ), where f is a common knowledge.
• More simplifying assumptions: - Each round every trader reports to the cen- tralized agent the E[ f ( x )| σ ], where σ is all the available information to that trader. - the centralized agent broadcasts the average value to all traders. - The market ”terminates” when there is no more trader can update E[ f ( x )| σ ]. - These assumptions may not be realistic.

Example 1: Two agents holding two bits x 1 and x 2 in the market. The function f ( x 1 , x 2 ) = x 1 ∨ x 2.

• Both agents know x 1 , x 2 are uniformly distributed.
• Agent 1 observed x 1 = 0 ; Agent 2 observed x 2 = 1.
• Agent 1 reports: E[ f ( x 1 , x 2 ) | x 1 = 0 ] = 0. 5.
• Agent 2 reports: E[ f ( x 1 , x 2 ) | x 2 = 1 ] = 1.
• Centralized agent received ( A 1 , 0. 5 ) and ( A 2 , 1 ) and broadcasts ( 0. 5 + 1 )/ 2 = 0. 75.
• Now based on knowing the average value 0. 75 , agents enters the 2nd round and provide their new estimate on the expectation of f ( x 1 , x 2 ).

Example 1: Two agents holding two bits x 1 and x 2 in the market. The function f ( x 1 , x 2 ) = x 1 ∨ x 2.

• Both agents know x 1 , x 2 are uniformly distributed.
• Agent 1 observed x 1 = 0 ; Agent 2 observed x 2 = 1.
• Agent 1 reports: E[ f ( x 1 , x 2 ) | x 1 = 0 ] = 0. 5.
• Agent 2 reports: E[ f ( x 1 , x 2 ) | x 2 = 1 ] = 1.
• Centralized agent received ( A 1 , 0. 5 ) and ( A 2 , 1 ) and broadcasts ( 0. 5 + 1 )/ 2 = 0. 75.
• Now based on knowing the average value 0. 75 , agents enters the 2nd round and provide their new estimate on the expectation of f ( x 1 , x 2 ).

Example 1: Two agents holding two bits x 1 and x 2 in the market. The function f ( x 1 , x 2 ) = x 1 ∨ x 2.

• Both agents know x 1 , x 2 are uniformly distributed.
• Agent 1 observed x 1 = 0 ; Agent 2 observed x 2 = 1.
• Agent 1 reports: E[ f ( x 1 , x 2 ) | x 1 = 0 ] = 0. 5.
• Agent 2 reports: E[ f ( x 1 , x 2 ) | x 2 = 1 ] = 1.
• Centralized agent received ( A 1 , 0. 5 ) and ( A 2 , 1 ) and broadcasts ( 0. 5 + 1 )/ 2 = 0. 75.
• Now based on knowing the average value 0. 75 , agents enters the 2nd round and provide their new estimate on the expectation of f ( x 1 , x 2 ).

History/Literature Review

• [PP82]: We cannot disagree forever.
• [MP86]: Common Knowledge, Consensus, and Aggregate Information - Provide a necessary condition on when all agents will finally “agree” on f ( x ). - But it is not always true that the converged value is the actual f ( x ).
• This paper: - Deal with discrete information and discrete func- tion. - Provided a necessary and sufficient condition for the agents’ estimate on E[ f ( x )] converges to f ( x ). - Provided upper bound for round complexity be- fore the process terminates.

Outline for the rest of the talk.

1. A few more examples.
2. Convergence result.
3. Necessary and sufficient condition for converging to the correct value.
4. Round complexity result.
5. Discussion on the open problems.
6. Relevant Followup work.

Outline for the rest of the talk.

1. A few more examples.
2. Convergence result.
3. Necessary and sufficient condition for converging to the correct value.
4. Round complexity result.
5. Discussion on the open problems.
6. Relevant Followup work.

Example 2. There are three agents holding 3 bits x 1 , x 2 , and x 3 , uniformly distributed. f ( x 1 , x 2 , x 3 ) is ma- jority function. i.e., f ( x 1 , x 2 , x 3 ) = I[ x 1 + x 2 + x 3 ≥ 2 ].

• x 1 = 1 , x 2 = 1 , and x 3 = 0.
• Round 1: Agent 1 reports 0.75, agent 2 reports 0.75, agent 3 reports 0.25.
• Centralized agent broadcast 0. 583.
• New common knowledge: { 110 , 011 , 101 }.
• Round 2: all agents know this common knowl- edge and output 1. The process terminates.

Example 2. There are three agents holding 3 bits x 1 , x 2 , and x 3 , uniformly distributed. f ( x 1 , x 2 , x 3 ) is ma- jority function. i.e., f ( x 1 , x 2 , x 3 ) = I[ x 1 + x 2 + x 3 ≥ 2 ].

• x 1 = 1 , x 2 = 1 , and x 3 = 0.
• Round 1: Agent 1 reports 0.75, agent 2 reports 0.75, agent 3 reports 0.25.
• Centralized agent broadcast 0. 583.
• New common knowledge: { 110 , 011 , 101 }.
• Round 2: all agents know this common knowl- edge and output 1. The process terminates.

Example 2. There are three agents holding 3 bits x 1 , x 2 , and x 3 , uniformly distributed. f ( x 1 , x 2 , x 3 ) is ma- jority function. i.e., f ( x 1 , x 2 , x 3 ) = I[ x 1 + x 2 + x 3 ≥ 2 ].

• x 1 = 1 , x 2 = 1 , and x 3 = 0.
• Round 1: Agent 1 reports 0.75, agent 2 reports 0.75, agent 3 reports 0.25.
• Centralized agent broadcast 0. 583.
• New common knowledge: { 110 , 011 , 101 }.
• Round 2: all agents know this common knowl- edge and output 1. The process terminates.

Example 3 Two agent 1 and 2 with private input x 1 and x 2. f ( x 1 , x 2 ) = x 1 ⊕ x 2. And x 1 , x 2 distributed uniformly.

1. x 1 = 1 and x 2 = 0.
2. For agent 1, he/she knows with half of the chance x 1 ⊕ x 2 is 1.
3. For agent 2, he/she knows with half of the chance x 1 ⊕ x 2 is 1.
4. They both output 0.5. The process terminates.