5.5 Evaluation
5.5.3 Solution Concept Success Rate
As detailed in Section5.1, if the DDP algorithms finds all coalitions that are explicitly detailed in the synergy coalitional group representation, then the payoff vector returned is guaranteed
to be in the weak least core+ of the superadditive cover (where the plus indicates that cross- coalition side payments are allowed). The DDP algorithms find every explicit coalition in the SCG representation in lines 7 to 15 of theCommunicatefunction (see Algorithm7), lines 6 and 20 of thedecodeDDP function (see Algorithm 9) and line 20 of theSplit function, if DDP∗is used (see Algorithm11). Then the linear program of theStablePayofffunction is solved for the grand coalition of the superadditive cover to find a weak least core+stable payoff vector of the superadditive cover.
If the core of the superadditive cover is non-empty then it will correspond to the weak least core+of the superadditive cover. Yet the success rate for the following solution concepts is not guaranteed using the DDP algorithm: (a) the least-strong-CS core+of the superadditive cover; or (b) the least-CS core. In [112], an example was detailed that showed that solutions within (b) maynotinclude the optimal coalition structure. This means that the DDP algorithms may not find solution (b).
An outcome is not guaranteed for strong least core+of the superadditive cover due to the in the strong-core definition (of Section2.2.1) not being proportional to the number of agents in the coalition and so a blocking coalition maybe missing from the SCG representation:
Theorem 5.6. Given a superadditive characteristic function game G = hN, vi and a payoff vectorx(wherex(N) = v(N)), letwbe the maximum weak excess that still gives a blocking coalition forxgiven full knowledge on each coalition’s value. Using only the coalition values inW, a blocking coalitionDfor the weak excess value ofs0is guaranteed to be present within W, wheres0is in the bound n−s1 ≤s0 ≤s.
Proof. Supposexis blocked by a coalition C throughv(C)−sso thatv(C)−s> x(C)and ∀s00 > s there is no blocking coalition. If(C, v(C)) ∈ W then this proves the Lemma. If
(C, v(C))∈/ W then from the definition of a SCG, it is known that the value of the coalition can be found through its maximum value partition, i.e.v(C)−s =P
1≤p≤qv(Cp)−s, for some set of coalitions{C1, ..., Cq}where:
1. Sq
p=1Cp =C
2. Ci∩Cj =∅for anyi, j∈ {1, ..., q}wherei6=j
3. (Cp, v(Cp))∈W, for allCp ∈ {C1, ..., Cq}
Via substitution, it follows thatP
1≤p≤qv(Cp)−s> x(C)and hence for at least oneCp then v(Cp)−
s
q > x(Cp). As the game is superadditive the grand coalition is known to form, hence the largest coalition that can have an excess value is S ⊂ N where |S| = n−1, therefore
0< q < n.
Using this information, a coalition with the strong excess value ofs0is guaranteed to be in the SCG representation, where: n−s1 ≤s0 ≤s. Thus the theorem is proved.
Therefore searching for the strong least core using only the coalitions in W may return erroneous results (i.e. a solution not in the strong least core) because the agents may conclude that a payoff vector is stable when inspecting the excess values of only the coalitions inW but
actually this payoff vector is not stable when inspecting the coalitions not inW. For further understanding, consider the following example:
Example 5.2. Consider a characteristic function game withN = {1,2,3,4}agents, where a coalitionC ={1,2,3}has a value ofv({1,2,3}) = 25. Given a payoff vectorx, that gives the coalitionCthe total valuex(C) = 22, it is known that the maximum (integer)strongexcess to give a blocking coalition iss0 = 2. For instancex(C) = 22< v(C)−s0 = 25−2 = 23, and so coalition{1,2,3}is a blocking coalition forxwhens0 = 2but not a blocking coalition whens0 = 3.
IfC ∈W then a blocking coalition for the maximum strong excess value (that still admits a blocking coalition) has been found. IfC /∈ W then there must be coalitions withinW that make up a partition ofCand have an equal or greater combined value thenC. In this example assume({1,2}, v({1,2}) = 16),({3}, v({3}) = 9)∈ W. The values of these coalitions have been chosen becausev({1,2}) +v({3}) =v({1,2,3}), i.e. the values are the minimal needed to make sure that coalition{1,2,3}is not in the SCG representation.
Given theses preliminaries, to stop coalition{1,2}blocking whens0 = 2, thenx({1,2})
must be greater than or equal tov({1,2})−s = 16−2 = 14. Assume thatx({1,2}) = 14
(i.e. the minimal payoff to satify the coalition has been given).
Recall x(C) = 22. Therefore x3 = x(C)−x({1,2}) = 22−14 = 8. But this gives x({3}) = 8> v({3})−s = 9−2 = 9−2 = 7. As the payoff of3is currently greater than its singleton value minuss0, the singleton coalition{3}does not become a blocking coalition for xunders0 = 2.
In conclusion this example shows that if a blocking coalition C with the maximum weak excess (that still admits a blocking coalition) isnotpresent in the SCG representation, then it is not guaranteed that a coalitionC0 ⊂C with the same weak excess value will be present in the SCG representation.
5.6
Summary
This chapter presented two decentralised algorithms, DDP and DDP∗, for agents to use to solve the coalition formation process in a completely distributed manner (i.e. without even shared memory access). Both algorithms are dynamic programming ones that are guaranteed to output a solution within the weak least core+ of the superadditive cover of the given characteristic function game, where the ‘+’ indicates that cross-coalition side payments are allowed. The algorithms distribute: (i) the coalition value calculations; (ii) the filtering of the coalitions into the synergy coalitional group representation; and (iii) the two-set partitions of each coalition to search. The agents communicate to each other information that allows each agent tonothave to compute all of (i), (ii) and (iii) by itself, yet the agents still achieve an optimal solution as if the individual agents had full knowledge.
The DDP and DDP∗algorithms have only one difference, which is that the DDP∗algorithm distributes the split operations of the grand coalition to the agents. This difference restricts the range of two-set partition operations required by the agents and leads to the interesting property,
proven in Theorem5.4, that whenever the number of agentsnis an odd prime, then the agents receive an exactly equal number of split operations to perform.
Both the DDP and DDP∗algorithms have been developed from the DP algorithm detailed in [138], as discussed in Section2.4.2. The DP algorithm only searched for the optimal coalition structure (referred to in [138] as the ‘best set partition’) and did not concern itself with stable payoff vectors. Another variation of DP is the IDP algorithm [91], which like DP finds the op- timal coalition structure, but by considering only 38.7% of the two-set partitions. Additionally, less memory was used for IDP compared to DP. The DDP algorithms followed the DP approach of considering all of the split partitions as this led to the interesting properties of: (a) being able to locate the exact coalitions in the synergy coalitional group representation; and (b) each agent having an equal number of split operations to consider when the agent number is an odd prime. Methods to reduce the memory requirements (described in [91]), such as removing thef2table through putting all operations forf2into thevtable, can easily be added to the DDP algorithms. Finally, the DDP algorithms use elements of the DCG algorithm of Chapter 4 because: (1) the DCG algorithm gives an approximately equal distribution for the agents coalition value calculation shares; and (2) every coalition in each agenti’s share has agentias a member. Point (2) is very important in the context of finding an accurate stable payoff vector distribution (in the context of complete knowledge), because without this point, then self-interested agents will be motivated to hide values of coalitions not including themselves in order to gain a larger payoff.
Valuation Disagreement Coalitional
Games
So far in Chapters 4 and 5, coalitional games with quantitative valuations were investigated where the values of the coalitions were the same for every agent, and therefore there were no valuation disagreements. In this chapter, coalitional games with valuation disagreements are investigated (where each agent assigns quantitative values to the coalitions). These disagree- ments can arise from agents having potentially different methods to interpret their possibly het- erogeneous knowledge bases, which can lead to the agents finding different conclusions on a coalition’s quantitative value. Thus finding stable coalitional game solutions in these situations involves identifying coalitions and agent-payoff contractual agreements that each agent finds acceptable given its own beliefs on each coalition’s quantitative value.
In this chapter, the newvaluation disagreement coalitional game(VCG) model is introduced with the new stability solution concepts of: the valuation-disagreement core, the valuation- disagreementv-coreand the non-emptyleast valuation-disagreement core. All of the solution concepts consist of a triple containing: (i) acoalition structure; (ii) anestimated payoff vector; and (iii) acontract function. The estimated payoff vector details each agenti’s expected payoff given all the agents’ coalition valuations, while the contract function provides an agreement on the manner to distribute the true value of each coalition in the coalition structure, once the true value of each coalition is revealed.
The new solution concepts allow harsh punishment of agents who overestimate the true value of their formed coalition in the agreed coalition structure. This harsh punishment is used to encourage agents to report lower pessimistic valuations. Pessimistic valuations can benefit an agent system as it increases the likelihood that each agent will gain a profit over its expected payoff once the true values of the formed coalitions are revealed. This is beneficial as less loss/debt will have to be assigned to some agent(s) from their expected payoffs (compared to the agents reporting normal or optimistic valuations). Assigning less debt is a very positive property due to coalitional games being a microeconomic model of the complicated human economic world, and assigning loss/debt can have a wide-ranging social impact on people (such as poverty, job losses, etc).
Therefore the contributions of this chapter are as follows: (i) the valuation disagreement coalitional gamemodel itself and its related solution concepts are introduced; (ii) these new so- lution concepts, when analysed empirically and compared quantitatively to coalitional games with beliefs1, allow previously unattainable coalitions to form; (iii) valuation disagreement coalitional games, compared tocoalitional games with beliefs, give solutions of greater expected value in the majority of cases (when the same valuations for both games are used); and (iv)val- uation disagreement coalitional game solution concepts are shown to encourage pessimistic valuations, compared to a solution concept that uses a single percentage-based agreement.
This chapter is structured as follows: in Section6.1, the valuation disagreement coalitional game model is introduced, which includes the estimated payoff vector and the associated sta- bility solution concepts. In Section6.2, the role of contract functions are described in detail. In Section6.3, an experimental evaluation is detailed. Finally Section6.4summarises and con- cludes.
6.1
The Proposed Solution
The motivation for this work can be found in the many possible methods to distribute unknown payoffs that can be identified in real economic environments where there is disagreement on a coalition’s true value, such as labour and financial markets. Perhaps the most popular method in coalitional game literature is that of Chalkiadakis et al. [33, 34] and Suijs et al. [122] who replace the traditional numeric payoff vector of characteristic function games with a fixed percentage-based demand vector. Yet this method is only a subset of the possible ways to dis- tribute the unknown value of a coalition. In these environments each agent has three main variables to consider before accepting or rejecting an offer to move to a new coalition:
1. Expected/estimated payoff and minimum payoff: Given agenti’s knowledge about the other agents of the coalition and the environment it is in, agentiestimates the worth of the coalitionC, denotedwi(C) in thevaluation disagreement coalitional game model. Given the agreed terms at the coalition valuationwi(C), agentican find itsestimated (or expected) payoff. The minimum payoff is the minimum estimated payoff an agent will accept to join a new coalition.
2. Share of profits: It is possible that all/some of the agent valuations could be an under- estimation of the true value. In this case, agents of the coalition will need an agreement to share any possible gains over all/some of their estimated payoffs.
3. Share of losses: Alternatively the agents may over-estimate all/some of their valuations of the coalitions. In this case, agents of the coalition will need an agreement to share any possible losses over all/some of their estimated payoffs. This is separate to variable 2 as some agents can make a profit on their estimated payoff when others make a loss.
1
Agents have different considerations for each of these variables. For instance, an agent iwill only consider joining a coalitionCif the estimated payoff that is being offered toiat the coali- tion valuewi(C)equals or exceeds agenti’s required minimum payoff. Each agentiwill also want to maximise its estimated payoff at the coalition valuewi(C). Additionally an agentimay be willing to sacrifice a percentage of payoff over a certain valuation ofCto an agentj(where this valuation would rationally be somewhere abovewi(C)), ifwj(C) is higher than wi(C). Sacrificing payoff at this higher valuation may entice agentj to join the coalition, possibly in return for more payoff under a valuation thatifinds more likely to occur.
Different approaches treat these three variables either all together (e.g. [33, 34, 122]), or exhaustively by creating a payoff vector for each different possible combination of coalition values (e.g. [65]). The method used in this chapter is different because it combines elements of the exhaustive payoff vector proposals [65] with those used in the single percentage demand vector proposals [33,34,122], as discussed in the next section.