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Bargaining in-Bundle over Multiple Issues in Finite-Horizon Alternating-Offers Protocol

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### 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 Ub (Us) of b (s) depends on her
- Reservation price RPb (RPs)
- Deadline Tb (Ts)
- Time discount factor δb (δs)
- Ub(x,t) = (RPb-x)(δb)t if t ≤ Tb
- Ub(x,t) = -1 if t > Tb
- Us(x,t) = (x-RPs)(δs)t if t ≤ Ts
- Us(x,t) = -1 if t > T
- Ub(NoAgreement) = Us(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 {Tb,Ts}
- 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=RPs or accept any offer x such that Ub(x,T-1) ≥ Ub(x*T-1,T)
- At time T-2 agent s would offer x*T-2 such that Ub(x*T-2,T-1) = Ub(RPs,T) or accept any offer x such that Us(x,T-2) ≥ Us(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

Ui(x*t-1,t) = Ui(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=<x*1t-1,…,x*nt-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:
- Ui(x1,…, xn,t) = ∑jUji(xj,t) if for each j Uji(xj,t) ≥ 0
- Ui(x1,…, xn,t) = -1 otherwise

where

- Uji(xj,t) = uji(xj)(δjb)t if t ≤ Tji
- Uji(xj,t) = -1 otherwise

where

- uji are continuous, concave and strictly monotonic
- uji are such that the agents have opposite preferences over each issue
- uji are such that there are feasible agreements

Multi-issue bargaining solution

- T = minji{Tji} is the global deadline of the bargaining
- Tuple x*T-1 = <x*1T-1,…,x*nT-1> = <RP1i,…,RPni> 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

Finally

Thank you for your kind attention

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