Bargaining in-Bundle over Multiple Issues in Finite- Horizon Alternating-Offers Protocol Francesco Di Giunta and Nicola Gatti Politecnico di Milano Milan, Italy

Summary Introduction to alternating-offers bargaining, open problems, and topic of the paper Review of the single-issue solution Basic ideas for our multi-issue solution Development of the multi-issue solution Conclusions and further work

Alternating-offers bargaining Two rational agents - a buyer b and a seller s – make offers and counteroffers in order to reach an agreement (e.g., on price, quality, quantity,… of a good to be sold) They have opposite interests and they both lose utility as time passes by Different settings: finite-horizon vs infinite-horizon single-issue vs multi-issue complete information vs incomplete information … The problem is: how should the two rational agents behave? Which should be their strategies?

Alternating-offers bargaining Game-theoretical analysis pioneered by [Stahl, 1972] and [Rubinstein, 1982] Long time interest in the game theory and in the artificial intelligence community The single issue problem with complete information is solved Slow further developments towards the solution of realistic models Main open problems: Incomplete information Multiple issues

Multi-issue problem Multi-issue bargaining protocols: Sequential: the issues are negotiated one by one In-bundle: all the issues are negotiated together Sequential bargaining does not assure Pareto- efficiency In-bundle bargaining is said to involve too much computations

Focus of our paper We focus on finite-horizon in-bundle alternating-offers bargaining with complete information We show that, for the most common kind of utility functions, the problem is indeed tractable We merge game-theoretical and linear/convex programming techniques

Review of the one-issue model The buyer b and the seller s act alternately at integer times Possible actions at time t are Make an offer (a real number, typically a price) Accept the opponent’s previous offer x: the outcome is (x,t) Exit the negotiation: the outcome is NoAgreement The utility function U b (U s ) of b (s) depends on her Reservation price RP b (RP s ) Deadline T b (T s ) Time discount factor δ b (δ s ) U b (x,t) = (RP b -x)(δ b ) t if t ≤ T b U b (x,t) = -1 if t > T b U s (x,t) = (x-RP s )(δ s ) t if t ≤ T s U s (x,t) = -1 if t > T U b (NoAgreement) = U s (NoAgreement) = 0

Review of the one-issue solution The appropriate notion of solution is subgame perfect Nash equilibrium The protocol is essentially a finite game, so the equilibrium can be found by backward induction: Call T = min {T b,T s } At time T the acting agent (say, s) would accept any offer with positive utility At time T-1 agent b would offer x * T-1 =RP s or accept any offer x such that U b (x,T-1) ≥ U b (x * T-1,T) At time T-2 agent s would offer x * T-2 such that U b (x * T-2,T-1) = U b (RP s,T) or accept any offer x such that U s (x,T-2) ≥ U s (x * T-2,T-1) … I.e., at each time point t, from T back, it is possible to recursively find the offer x * t that the acting rational agent would do if she would make an offer; such offers x * t (or possible irrational higher ones) are always accepted by the rational opponent. Therefore the agreement is achieved at the very beginning of the bargaining on the value x * 0

Towards the multi-issue solution The core of the single-issue solution is the calculation of the values x * t that one agent should offer at time t and the other should accept at time t+1 In the one-issue situation this is very easy Are there, in the multi-issue situation, tuples x * t of values that act somehow like these values x * t ? The answer, for a wide class of multi-issue utility functions, turns out to be yes Is the calculation of these values computationally tractable? Again, the answer is yes Is the attained agreement Pareto-efficient? Yes

Towards the multi-issue solution In single-issue bargaining, value x * t-1 is calculated from x * t as the value such that U i (x * t-1,t) = U i (x * t,t+1) where i is the agent that acts at time t I.e., x * t-1 is obtained as the one step “backward propagation” of x * t along the level curves of the utility function of agent i In multi-issue bargaining, instead, there is no unique “backward propagated” tuple x * t-1 = but an entire set of tuples X * t-1 which at time t are worth for agent i the same as x * t at time t+1

Basic idea for multi-issue bargaining We take as x * t-1 the tuple in X * t-1 that maximizes the utility of the agent acting at time t-1 For a wide range of utility functions, this can be done efficiently with linear/convex programming.

Multi-issue bargaining assumptions Linear multi-issue utility function of agent i: U i (x 1,…, x n,t) = ∑ j U j i (x j,t) if for each j U j i (x j,t) ≥ 0 U i (x 1,…, x n,t) = -1 otherwise where U j i (x j,t) = u j i (x j )(δ j b ) t if t ≤ T j i U j i (x j,t) = -1 otherwise where u j i are continuous, concave and strictly monotonic u j i are such that the agents have opposite preferences over each issue u j i are such that there are feasible agreements

Multi-issue bargaining solution T = min ji {T j i } is the global deadline of the bargaining Tuple x * T-1 = = where i is the agent that acts at time T To calculate x * t-1 from x * t (be s the agent that acts at time t) Calculate the set X * t-1 of tuples which at time t are worth the same as x * t at time t+1 for agent s Use linear/convex programming to calculate x * t-1 as the value in X * t-1 that maximizes the utility of agent b

Multi-issue bargaining solution Be σ* the following strategy profile: At time T accept any offer that has nonnegative value At time t<T accept any offer x such that agreement (x,t) has utility greater or equal to (x * t-1,t+1) and otherwise counteroffer x * t

Main results It can be shown that Strategy σ* is the unique subgame perfect equilibrium of the protocol The calculation of σ* is linear with T and polynomial with the number of issues With strategy profile σ*, the agreement is achieved immediately and is Pareto-efficient

Conclusions In this paper we have shown that complete information multi-issue bargaining is tractable, despite what is usually believed, for a wide (and the most common) range of utility functions and for the best known bargaining protocol Further work will deal with the incomplete information problem

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