Carleman's condition

The Zeuthen strategy is a negotiation strategy used by some artificial agents. Its purpose is to measure the willingness to risk conflict. An agent will be more willing to risk conflict if the difference in utility between its current proposal and the conflict deal is low.

When used by both agents in the Monotonic Concession Protocol, the Zeuthen strategy leads the agents to agree upon the deal in the negotiation set, the set of all conflict free deals which are Individually rational and Pareto optimal plus the conflict deal, which maximizes the Nash product.

Risk

$\text{Risk}(i,t)=\{\begin{array}{ll}1& U_i(\delta (i,t))=0\\ \frac{U_i(\delta (i,t))-U_i(\delta (j,t))}{U_i(\delta (i,t))}& \text{otherwise}\end{array}$

Risk(A,t) is a measurement of agent A's willingness to risk conflict. The risk function formalizes the notion that an agent's willingness to risk conflict is the ratio of the utility that agent would lose by accepting the other agent's proposal to the utility that agent would lose by causing a conflict. Agent A is said to be using a rational negotiation strategy if at any step t + 1 that agent A sticks to his last proposal, Risk(A,t) > Risk(B,t).

Sufficient concession

If agent A makes a sufficient concession in the next step, then, assuming that agent B is using a rational negotiation strategy, if agent B does not concede in the next step, he must do so in the step after that. The set of all sufficient concessions of agent A at step t is denoted SC(A, t).

Minimal sufficient concession

${\delta }^{\prime }=\arg {\max }_{\delta \in SC(A,t)}\{U_A(\delta )\}$

is the minimal sufficient concession of agent A in step t.

Zeuthen strategy

Agent A begins the negotiation by proposing

$\delta (A,0)=\arg {\max }_{\delta \in NS}{U}_A(\delta )$

and will make the minimal sufficient concession in step t+1 if and only if Risk(A,t) ≤ Risk(B,t).

Theorem If both agents are using Zeuthen strategies, then they will agree on

$\delta =\arg {\max }_{{\delta }^{\prime }\in NS}\{\pi ({\delta }^{\prime })\},$

that is, the deal which maximizes the Nash product. (Harsanyi 56)

Proof Let δ_A = δ(A,t). Let δ_B = δ(B,t). According to the Zeuthen strategy, agent A will concede at step $t$ if and only if

$Risk(A,t)\le Risk(B,t).$

That is, if and only if

$\frac{U_A({\delta }_A)-U_A({\delta }_B)}{U_A({\delta }_A)}\le \frac{U_B({\delta }_B)-U_B({\delta }_A)}{U_B({\delta }_B)}$

$U_B({\delta }_B)(U_A({\delta }_A)-U_A({\delta }_B))\le U_A({\delta }_A)(U_B({\delta }_B)-U_B({\delta }_A))$

$U_A({\delta }_A)U_B({\delta }_B)-U_A({\delta }_B)U_B({\delta }_B)\le U_A({\delta }_A)U_B({\delta }_B)-U_A({\delta }_A)U_B({\delta }_A)$

$-U_A({\delta }_B)U_B({\delta }_B)\le -U_A({\delta }_A)U_B({\delta }_A)$

$U_A({\delta }_A)U_B({\delta }_A)\le U_A({\delta }_B)U_B({\delta }_B)$

$\pi ({\delta }_A)\le \pi ({\delta }_B)$

Thus, Agent A will concede if and only if ${\delta }_A$ does not yield the larger product of utilities.

Therefore, the Zeuthen strategy guarantees a final agreement that maximizes the Nash Product.

References

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