ComunicaciónComunicación
6.1 Comunicación corporativa
The valuation disagreement coalitional game (VCG) model provides more flexibility in coali- tional agreements than [33,34,122] by allowing different percentage distributions for the dif- ferent possible true values of each coalition. Yet the VCG model does not state an explicit agreement for every possible true value, due to this requirement being extremely complex even for situations with small amounts of valuation disagreement [65]. Instead, VCG relies on a contract function that gives an implicit payoff agreement for every possible true value.
The VCG does not state where exactly the difference in each agent’s opinion on a coalition’s value comes from (be it from agent types, possible worlds, or another method). Yet this chapter’s model, like the coalitional games with beliefs model of [34], uses the simplifying assumption of single point beliefs for the reasons stated in Section2.3.2. These reasons are: probabilistic beliefs may not be available; single point beliefs are easier to form; reasoning over single point beliefs is computationally less complex; and single point beliefs are a natural assumption in many real-world situations. Given these preliminaries, valuation disagreement coalitional games are defined as:
Definition 83: An valuation disagreement coalitional game is: Gv = (N;w) where N =
{1, .., n} is the set of agents andw is the valuation vectordenoted w = hw1, ..., wni where eachwi ∈ w assigns every possible coalition a value (wi(2N) → R) representing agent i’s
reported best guess on each coalition’s value. The notationw(C)denotes all the valuations of Cby the agents ofC.
In this chapter, it is assumed that each agent reportswiso that it is public knowledge. Then the valuation disagreement coalitional game outcome can accurately reflect each agent’s re- ported valuations (which might not be their truthful valuations). The justification of this can be seen in many real life negotiations where the future value of a coalition is disputed and individuals make known their valuations to improve negotiation. For example, in the popu- lar international television programme “Dragons’ Den”, where business ideas are pitched to a
panel of venture capitalists, the two or more negotiating parties usually make known their fu- ture valuation of the business when demanding a percentage share (of future payoff) to form the coalition.
Reporting personal information can create strategic issues. In this chapter, unlike [78], the concern is not with designingincentive compatiblemechanisms where the agents perform best if they truthfully reveal their private information. Finding an incentive compatible mechanism can be a computationally difficult task [119].
Instead, like [23], in this chapter it is assume that the agents report coalition valuations in anear incentive compatible manner; that is, the agents cannot determine how to change their coalition valuations (from their truthful valuations) to increase their final payoff, even though this maybe theoretically possible.
A simple way to ensure near incentive compatibility is for each agent i ∈ N to have no knowledge on the other agent’s coalition valuations whenireports its coalition values. To do this, each agentishould encrypt its valuations, send them to every other agentj(i.e. j∈N,j6=
i), and for the decryption key to be sent only whenihas received every other agentj’s encrypted valuations [23]. Now in this situation, each agentiwill not know if raising a single coalition’s value (above the coalition’s truthful value) will giveimore power to demand more payoff from some other agentsorcause another coalition to become unstable and dissolve, thus decreasing the total possible payoff agentican demand from other agents anyway. If each agent’s coalition valuations were exchanged like this, then the agents would have complete knowledge on each agent’s coalition valuations and could continue to reason over which coalitions to form via coalitional bargaining, a payoff vector transfer scheme or another type of distributed algorithm (for example, similar to the previous chapter’s distributed algorithm).
To reason over outcomes of valuation disagreement coalitional games, it is useful to intro- duce a functionπ(C, i) that takes as input a coalition C and an agenti ∈ C and outputs all the agentsM ⊆ C (where i ∈ M), who believe the value ofC to be equal to or less than wi(C), i.e. for allj ∈M,wj(C)≤wi(C)holds. Using this function, estimated payoff vectors that are used to find acceptable outcomes for valuation disagreement coalitional games can be introduced, formalised as:
Definition 84: Anestimated payoff vector denotede = he1, ..., enifor a coalition structure CS ={C1, ..., Ck}in a gameGv satisfies: (a)ei ≥wi({i})(i.e.individual rationality) for all i∈N; and (b)P
j∈π(C,i)ej ≤wi(C)for each agenti∈N, wherei∈C ∈CS. The estimated payoff vector of a coalitionC is denotede(C). Usinge, the expected payoff for each agentiis given byei.
Estimated payoff vectors give the agents more flexibility when negotiating for payoff for a coalition C, as each agentionly has to offer a distribution of wi(C) that satisfies all agents M ⊆Cwho valueC less than or equal toi. The total estimated payoff to theM agents given byπ(C, i)must never be greater thanwi(C)for any coalitionC ∈CS, which ensures that each agentican achieve its estimated valuation should the coalition’s real value be equal to or greater thanwi(C).
0 5 10 15 0 2 4 6 8 10
Possible Coalition Value
Agent P ayof f ag1 ag2
FIGURE 6.1: This figure shows how a simple contract changes agentag1 andag2’s payoff given different possible future valuations of the coalition {ag1, ag2}, whereag1 requires a payoff of 4 (or more) when the coalition is valued at 5 (or more) andag2requires a payoff of 6
(or more) when the coalition is valued at 10 (or more).
The logic behind the estimated payoff vectors is simple. Consider two agents ag1 andag2 negotiating over possibly joining together in a coalition. Agentag1reports the coalition’s value as 5 and requires an estimated payoff of 4 to join (i.e. 80% of 5). Agent ag2 reports the coalition’s value as 10 and requires an estimated payoff of 6 to join (i.e. 60% of 10). Then both agents will be satisfied (if the agents are risk neutral) with a contract that gives agent ag1 a monotonically increasing payoff that will be≥4 at the valuation 5 and gives agentag2 a monotonically increasing payoff that will be ≥ 6 at the valuation of 10. Additionally this contract needs to state how to work out each agent’s payoff for any possible true valuation of the coalition. A simple contract (i.e. one of many possible contracts), such as the one presented in Figure6.1, could assign all the payoff to the agent with the smallest valuation (ag1) until its demand is reached, then all the payoff to the agent with the next smallest valuation (ag2) until its demand is reached, and finally assign any payoff, over the total of both demands, proportionally to both agents.
These contracts are represented via a functionκthat takes as input a tuplehC, e(C), w(C), v(C)i representing: the coalition (i.e.C); the estimated payoff of the agents in the coalition (i.e.e(C)); the reported agent valuations for that coalition (i.e. w(C)); and the coalition’s true value (i.e. v(C)). Theκfunction outputs a payoff vector agreementx(C)for the coalition at its true value v(C).
More complicated contracts can be developed that distribute any possible future valuation of the coalition. These contracts could be designed for different purposes (or a combination of different purposes), such as: (i) encouraging conservative valuations; (ii) distributing the payoff in the most ‘fair’ manner; and (iii) distributing the payoff proportionally to some variable. The two restrictions assumed in contract functions (in this thesis) are:
1. Each agent’s payoff is monotonically weakly increasing as the coalition value increases.
2. Each agentimust receive equal to or greater than its estimated payoffeiwhen the coali- tionCis formed andv(C)≥wi(C).
Restriction 1 means a situation cannot occur where an agent gets a payoff ofyifv(C) =z and a payoff of less thany ifv(C) > z. Restriction 2 means that it is guaranteed that if any agent icorrectly predicts or pessimistically values the coalition it forms, then iis to be given greater than or equal to its estimated payoffei.
Now that the estimated payoff vector and the contract function have been introduced, the outcome of a valuation disagreement coalitional game can be defined:
Definition 85:Anoutcomeof a valuation disagreement coalitional game is a triple consisting of a coalition structure, an estimated payoff vector and a contract function, denoted:hCS, e, κi.
The different possible contract functions do not affect valuation disagreement coalitional game stability solution concepts (introduced in the next sub-section), yet may influence what valuations each agent reports. The valuation disagreement coalitional game stability solution concepts are not affected by the contract functions, because these concepts are only concerned with finding a stable outcome given the reported valuations. The contract functions are only used if the highest valuation of any coalition in the final coalition structure is wrong. In this case, some more (resp. less) payoff needs to be assigned to (resp. from) the coalition’s members as the coalition’s true value is higher (resp. lower) than expected.