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### Distributed Rational Decision Making

Author: Tuomas W. Sandholm

Speakers: Praveen Guddeti (1---5)

Tibor Moldovan (6---9)

CSE 976, April 15, 2002

Outline

- Introduction
- Evaluation criteria
- Non-cooperative interaction protocols
- Voting
- Auctions
- Bargaining
- General equilibrium market mechanisms
- Contract nets
- Coalition formation
- Conclusions

Introduction

- Automated negotiation systems with self-interested agents are becoming increasing important.

1. Technology push.

2. Application pull.

- Paper deals with protocols designed using a non-cooperative, strategic perspective.

Evaluation Criteria

- Social welfare
- Pareto efficiency
- Individual rationality
- Stability
- Computational efficiency
- Distribution and communication efficiency

Evaluation Criteria1. Social Welfare

- It is the sum of all agents’ payoffs or utilities in a given solution.
- Requires inter-agent utility comparisons.

Evaluation Criteria2. Pareto Efficiency

- A solution x is Pareto efficient if there is no other solution x’ such that
- at least one agent is better off in x’ than in x, and
- no agent is worse off in x’ than in x.
- Does not require inter-agent utility comparisons.
- Social welfare maximizing solutions are a subset of Pareto efficient ones.

Evaluation Criteria3. Individual Rationality

- Participation in a negotiation is individually rational to an agent only if it is profitable.
- A mechanism is individually rational if participation is individually rational for all agents.
- Only individually rational mechanisms are viable.

Evaluation Criteria4. Stability

- The protocol mechanisms should motivate each agent to behave in the desired manner.
- Protocol mechanisms may have dominant strategies. This means that an agent is best off by using a specific strategy no matter what strategies the other agents use.
- Nash equilibrium: Each agent chooses a strategy that is a best response to the other agents’ strategies.

Nash Equilibrium:Formal definition

- The strategy profile S*A={ S*1, S*2,…, S*|A|} among agents A is in Nash equilibrium if for each agent i , S*i is the agent’s best strategy given that the other agents choose strategies {S*1, S*2,…, S*i-1, S*i+1,… S*|A|}.
- In some games no Nash equilibrium exists.
- Some games have multiple Nash equilibrium.

Nash Equilibrium:Comments

- Even if Nash equilibrium exists and is unique, there are limitations regarding what the Nash equilibrium guarantees.
- In sequential games it only guarantees stability in the beginning of the game.
- Subgame perfect Nash equilibrium.
- Nash equilibrium is often too weak because subgroups of agents can deviate in a coordinated manner.
- Sometimes efficiency and stability goals conflict.

Evaluation Criteria5. Computational Efficiency

- The protocol mechanisms when used by agents should need as little computation as possible.
- Trade off between:
- the cost of the computation needed for the protocol mechanisms and
- the solution quality.

Evaluation Criteria6. Distribution and Communication Efficiency

- Distributed protocols should be preferred in order to avoid a single point of failure and a performance bottleneck – among other reasons.
- Minimize the amount of communication required to get to a desired global solution.
- These two goals can conflict.

Non-cooperative interaction protocols

- Voting
- Auctions
- Bargaining
- General equilibrium market mechanisms
- Contract nets
- Coalition formation

Non-cooperative Interaction Protocols1. Voting

- All agents give input to a mechanism.
- Outcome chosen by the mechanism is solution for all agents.
- Outcome is enforced.
- Voters.
- Truthful voters.
- Strategic (Insincere) voters.

VotersTruthful Voters (1)

- Each agent i A has an asymmetric and transitive strict preference relations i on O.
- Social choice rule.
- input the agents’ preference relations ( 1,…, |A|).
- output the social preferences denoted by a relation *.

Truthful Voters (2)Properties of a social choice rule

- * should exist for all possible inputs.
- * should be defined for every pair o, o’ O.
- * should be asymmetric and transitive over O.
- The outcome should be Pareto efficient.
- The scheme should be independent of irrelevant alternatives.
- No agent should be a dictator.

Truthful Voters (3)

- Arrow’s Impossibility Theorem: No social rule satisfies all of these six conditions.
- Relax the first property.
- Relax the third property.
- Plurality protocol.
- Binary protocol.
- Borda protocol.

Truthful Voters (4)Plurality Protocol

- Majority voting protocol.
- All alternatives are compared simultaneously.
- The one having the highest number of votes wins.
- Irrelevant alternative can split the majority.

Truthful Voters (5)Binary Protocol (1)

- Pair wise voting with the winner staying to challenge remaining alternatives.
- Irrelevant alternatives can change outcomes.
- Agenda i.e. order of the pairings can change the outcomes.

Truthful Voters (6)Binary Protocol (2)

35 % of agents have preferences cdba

33 % of agents have preferences acdb

32 % of agents have preferences bacd

Truthful Voters (7)Borda Protocol (1)

- Assign an alternative |O| points whenever it is the highest in some agent’s preference list, |O| -1 when it is second and so on.
- Sum the counts of all alternatives.
- Alternative with highest count is the winner.
- Irrelevant alternatives lead to paradoxical results.

Borda Protocol (2)

Agent Preferences

1 a b c d

2 b c d a

3 c d a b

4 a b c d

5 b c d a

6 c d a b

7 a b c d

Borda count c wins with 20, b has 19, a has 18, d loses with 13

Borda count a wins with 15, b has 14,loses with 13

with d removed

VotersStrategic (Insincere) Voters (1)

- Revelation principle: Suppose some protocol implements social choice function f(.) in Nash (or dominant strategy) equilibrium, then f(.) is implementable in Nash (or dominant strategy) equilibrium via a single-step protocol where the agents reveal their types truthfully.

VotersStrategic (Insincere) Voters (2)

- Gibbard-Satterthwaite impossibility theorem: Let each agent’s type i, consist of a preference order i on O. Let there be no restrictions on i, i.e. each agent may rank the outcomes O in any order. Let |O| 3. Now, if the social choice function f(.) is truthfully implementable in a dominant strategy equilibrium, then f(.) is dictatorial, i.e. there is some agent i who gets (one of) his most preferred outcomes chosen no matter what types the others reveal.

VotersStrategic (Insincere) Voters (3)

- Circumventing the GSIT:
- Restricted preferences.
- Groves-Clarke Tax Mechanism.
- Groves-Clarke Tax Mechanism:
- o = (g ,1,… |A|).
- i is the amount agent i receives.
- g encodes the other features of the outcome.

Strategic (Insincere) Voters (4)Groves-Clarke Tax Mechanism(1)

- Quasilinear preferences: ui(o) = vi(g) + i.
- Net benefit: vi(g) = vi gross(g) – P / |A|.
- Every agent iA reveals his valuation vi(g) for every possible g.
- The social choice is g* =arg maxg i vi(g).
- Every agent is levied a tax: tax i = ji vj (g*) - ji vj (arg maxg ki vk (g)).

Strategic (Insincere) Voters (5)Groves-Clarke Tax Mechanism(2)

- Size of an agent’s tax is exactly how much his vote lowers the other’s utility.
- Quasilinearity:
- No agent should care how others divide payoffs among themselves.
- An agent’s valuation vi gross(g) should not depend on the amount of money that the agent will have.

VotersStrategic (Insincere) Voters (6)

- If each agent has quasilinear preferences, then each agent’s dominant strategy is to reveal his true preferences.
- Agents need not waste effort in counter speculating each others’ preference declarations.
- Participation is individually rational.

VotersStrategic (Insincere) Voters (7)

- Problems of Groves-Clarke Tax Mechanism:
- Does not maintain budget balance.
- Not coalition proof.
- Intractable.
- Other ways to circumvent the GSIT:
- Choosing a dictator randomly.
- Make the computation of an untruthful revelation prohibitively costly.

Non-cooperative Interaction Protocols2.Auctions

- Unlike voting where the outcome binds all agents, in auctions the outcome is usually a deal between two agents.
- In voting the protocol designer wants to enhance the social good, while in auctions, the auctioneer wants to maximize his own profit.
- Classical setting.
- Contracting setting.

Auctions (2)

- Auction settings.
- Auctions protocols.
- Efficiency of the resulting allocation.
- Revenue equivalence and non-equivalence.
- Bidder collusion.
- Lying auctioneer.
- Bidders lying in non-private-value auctions.
- Undesirable private information revelation.
- Roles of computation in auctions.

1. Auction settings

- Three qualitatively different auctions depending on how an agent’s value of the item is formed:
- Private value.
- Common value.
- Correlated value.

2. Auctions protocols

- English (first-price open-cry) auction.
- First-price sealed-bid auction.
- Dutch (descending) auction.
- Vickrey (second-price sealed-bid) auction.
- Allocation of computation resources in OS, allocation of bandwidth in computer networks, computationally control building heating.
- Has not been widely adopted in auctions among humans.

3. Efficiency of the resulting allocation.

- In isolated private value or common value auctions, each one of the four auction protocols allocates the auctioned item Pareto efficiently to the bidder who values it the most.
- All four protocols are Pareto efficient in the allocation.
- The dominant strategies (Vickrey and English) are more efficient.

4. Revenue equivalence and non-equivalence.

- Revenue equivalence: All of the four auction protocols produce the same expected revenue to the auctioneer in private value auctions where the values are independently distributed and bidders are risk-neutral.
- Among risk averse bidders, the Dutch and the first-price sealed-bid protocols give higher expected revenue to the auctioneer.
- A risk averse auctioneer achieves higher expected utility via the Vickrey or English protocols.

Revenue equivalence and non-equivalence. (2)

- In non-private value auctions, both the English and Vickrey protocols produce greater expected revenue to the auctioneer than the first-price sealed-bid auction or Dutch auction.
- In non-private value auctions with at least three bidders, the English auction leads to higher revenue than the Vickrey auction.

5. Bidder collusion.

- The English auction and the Vickrey auction actually self-enforce some of the most likely collusion agreements.
- First-price sealed-bid and the Dutch auctions are preferred for deterring collusion.
- For collusion to take place in Vickrey, first-price sealed-bid or Dutch auctions the bidders have to identify each other before placing the bids.

6. Lying Auctioneer.

- In Vickrey auction the auctioneer may lie about the value of the second highest bidder.
- In the English auction the auctioneer can use shills that bid in the auction in order to make the real bidders increase their valuations of the item.
- The auctioneer may bid himself to guarantee that the item will not be sold below a certain price.

7. Bidders lying in non-private-value auctions.

- Winner’s curse: If an agent bids its valuations and wins the auction, it will know that its valuation was too high because the other agents bid less.
- Agents should bid less than their valuations.
- This is the best strategy in Vickrey auctions.
- Vickrey fails to induce truthful bidding in most auction settings.

8. Undesirable private information revelation.

- In Vickrey auctions the agents often bid truthfully. This leads to the bidders revealing their true valuations.
- This information is sensitive and the bidders would prefer not to reveal it.
- Another reason why the Vickrey auction protocol is not widely used among humans.

9. Roles of computation in auctions.

- Two issues arise from computation in auctions:
- Computationally complex look ahead that arises when auctioning interrelated items one at a time.
- Implications of costly local marginal cost (valuation) computation or information gathering in a single-shot auction.

Interrelated auctions.

- Look ahead:
- Without look ahead the allocation may be inefficient.
- With look ahead the agents will not bid their true per-item cost.
- Computation cost may be prohibitively great.
- Allow agents to backtrack from commitments by paying penalties.

Single-shot auctions

- Incentive to counter speculate: In a single-shot private value Vickrey auction with uncertainty about an agent’s own valuations, a risk neutral agent’s best action can depend on the other agents. It follows that is is worth counter speculating.

Non-cooperative Interaction Protocols3. Bargaining

- Real world settings usually consist of a finite number of competing agents, so neither monopoly,nor monopsony nor perfect competition assumptions strictly apply.
- Bargaining theory fits in this gap.
- Bargaining theory:
- Axiomatic
- Strategic

Bargaining Theory Axiomatic

- Does not use the idea of equilibrium.
- Desirable properties for a solution, called axioms of the bargaining solution, are postulated.
- Then the solution that satisfies these axioms are sought.
- Nash bargaining solution.

Axiomatic Bargaining Theory Nash Bargaining Solution (1)

- Nash analyzed a 2-agent setting where the agents have to decide on an outcome o O, and the fallback outcome ofallback occurs if no agreement is reached.
- There is a utility function

ui: O R for each agent i [1,2].

- It is assumed that the set of feasible utility vectors { (u1 (o), u2 (o)) | o O} is convex.

Axiomatic Bargaining TheoryNash Bargaining Solution (2)

- Axioms for the Nash bargaining solution u* = (u1(o*), u2(o*)) are:

1. Invariance.

2. Anonymity (symmetry).

3. Independence of irrelevant alternatives.

4. Pareto efficiency.

Axiomatic Bargaining TheoryNash Bargaining Solution (3)

- The unique solution that satisfies these four axioms is:

o* = arg maxo[u1(o)–u1(ofallback)][u2(o)–u2 (ofallback)]

- Other bargaining solutions also exist.

Bargaining TheoryStrategic (1)

- Bargaining situation is modeled as a game.
- Solution is based on an analysis of which of the players’ strategies are in equilibrium.
- Solution is not unique.
- Explains the behavior of rational utility maximizing agents better than axiomatic approaches.
- Usually analyses sequential bargaining.

Bargaining TheoryStrategic (2)

- Finite number of offers with no time discount.
- Finite number of offers with time discount.
- Infinite number of offers with no time discount.
- Infinite number of offers with time discount.

Strategic Bargaining TheoryRubinstein Bargaining Solution

- In a discounted infinite round setting, the subgame perfect Nash equilibrium outcome is unique. Agent 1 gets (1- 2) / (1- 12), where 1 is 1’s discount factor, and 2 is 2’s. Agent 2 gets one minus this. Agreement is reached in the first round.
- The proof gives a way to solve for subgame perfect Nash equilibrium payoffs.

Strategic Bargaining TheoryFixed Bargaining Cost per negotiation round

- If the agents have symmetric bargaining costs, the solution concept is powerless.
- If 1’s bargaining cost c1 is even slightly smaller than 2’s cost c2, then 1 gets the entire dollar.
- If 1’s bargaining cost is greater than 2’s, then 1 receives a payoff that equals the second agent’s bargaining cost, and agent 2 receives the rest.

Strategic Bargaining TheoryRecent extensions [Kraus et al.]

- Sequential bargaining with outside options.
- Sequential bargaining where one agent gains and one loses over time.
- Negotiation over time when agents do not know each other’ types.

BargainingComputation (1)

- Assume perfect rationality.
- The space of deals is assumed to be fully comprehended by the agents.
- The value of each potential contract known.
- Focus of future work:
- make the cost of search explicit and.
- consider its trade-off with bargaining gains.

BargainingComputation (2)

- There are two searches occurring in bargaining:
- Intra-agent deliberative search: an agent locally generates alternatives, evaluates them, counter speculates, does look ahead etc.
- Inter-agent committal search: the agents make agreements with each other regarding the solution.

Introduction

- Evaluation criteria
- Non-cooperative interaction protocols
- Voting
- Auctions
- Bargaining
- General equilibrium market mechanisms
- Contract nets
- Coalition formation
- Conclusions

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