The following is a discussion on how the DCG algorithm presented in this chapter, compares to the DCVC algorithm [89,90], the VBFR algorithm [126] and the SK algorithm [117], when judged on the following properties identified in the introduction of this chapter: (i) elimination of communication; (ii) elimination of coaltion value calculation redundancy; (iii) approximately equal coalition value calculation shares for each agent; (iv) agents with equal sized shares have the same number of operations to perform; and (v) each coalition in an agent’s share includes that agent as a member.
(i) Elimination of communication: The method to split up the calculations of the coalition values in the DCG algorithm has no communication costs (as you can see there is no where in the DCG algorithm that requires communication between the agents). No communica- tion costs occur because, like the DCVC and VBFR algorithms, the agents have: (1) a pre-agreed ordering of the coalitions; and (2) a pre-agreed algorithm to dictate which ex- act share of this ordering each agent should calculate. Conditions (1) and (2) are not the case for the SK algorithm and so agents using this algorithm need to communicate to find out which coalitions will be in each agent’s share. Yet for any distributed coalition value calculation algorithm, if the agents want to complete the decentralised coalition formation process, communication costs will be incurred later when agents communicate to each other their best coalitions or coalition structures found (e.g. [1,73,104]). Even when the best coalition structure is known, more communication will occur if the agents want to negotiate on their final utility payoff (e.g. [1,7, 30,59,67, 118,137]). How the agents can communicate the correct coalitions (given coalition shares according to the DCG algo- rithm), constrain communication costs and achieve a certain stable outcome is described in Chapter5.
(ii) Elimination of Redundancy: In the DCG algorithm: (1) one and only one representative increment array from each equivalence class is used. This single representative increment
arraytis only used: (2)(n×π(t))/stimes; and (3) no more than once for each agent. In AppendixAit is proved that a repeated coalition can only be generated from an increment array if any of (1), (2) and (3) are not abided to. The DCVC algorithm and VBFR algo- rithm also do not have any redundant coalition value calculation issues because the agents are aware how the coalitions are ordered in both algorithms and therefore each agent’s share does not accidentally overlap. The SK algorithm does involve an exponentially large redundancy because, as stated in [90], each agent commits to calculate the value of a set of coalitions with limited knowledge on the other agent’s commitments.
(iii) Approximately equal coalition value calculation shares: The DCG algorithm, like the DCVC algorithm, gives approximately equal shares. The DCG algorithm has a maximum difference of two between the agent’s coalition value calculation share due to the each agent generating 2 coalitions at a time. The DCG algorithm achieves this through the use of thebalpointer, while the DCVC algorithm uses theα pointer (where theα pointer is described in Section2.4.1).
Thebalpointer is initialised to 1 in line 4 of the DCG algorithm. Thenballets the nextr agents calculate a maximum of two coalition values, after that, balis corrected (line 21). Asbalonly allows a maximum of two coalitions to be calculated at a time, an agent can only be a maximum of two coalition assignments ahead of another agent.
The VBFR algorithm does not give balanced shares to the agents (in the case where all the agents are assumed to be in a fully connected graph). In this case, fornagents, agent 1 is always assigned2n/2coalitions because there are2n/2coalitions where agent ID 1 is the smallest ID of that coalition. Agent n on the other hand is always assigned only one coalition because there is only one coalition with agentnas the smallest ID, which is the singleton coalition{n}. Finally the SK algorithm has no guarantees on the maximum difference between the agent’s shares, yet the average difference grows exponentially with the number of agents, as detailed in [90].
(iv) Equal number of operations for agents with equal sized shares: Only the DCG algo- rithm and the DCVC algorithms guarantee that the agents coalition value calculation shares will be approximately equal. Yet [90] shows that the DCVC algorithms do not guarantee an equal number of operations5to generate each agent’s coalition value calculation share, even when each agent’s share is equal sized. This is due to the lexicographical ordering of the coalitions that the DCVC algorithm uses. For the DCG algorithm equal operations will be performed by agents that are allocated an equal number of coalitions of each size, as each IA of the same size requires the exact same number of operations of additions to find the corresponding coalition (see Definition 79). This is because the DCG algorithm, unlike the DCVC algorithm, does not rely on lexicographical order to generate the coalitions. In- stead two-dimensional lists based on increment arrays are used. For every increment array of sizes, an agent needs to uses−1additions to generate the associated coalition, even if the increment arrays are different. The increment arrays used by the agents are the only
5 10 15 20 0 10 20 30 40 50 60 70 80 90 100 Number of Agents The % of Coalitions including the Calculating Age nt DCVC DCVC2 DCVC3 DCG
FIGURE4.1: This graph shows, for annagent game, the percentage of coalitions in allCVi∈N that include agenti(for all sizes1≤s≤n). It was produced from a java implementation of all the algorithms, where each agent kept a count of each coalition assigned to itself that included
itself.
variable points of the DCG algorithm. Therefore for any two agents that generate the same number of coalitions for everys, then the same number of increment arrays are used, thus the same number of operations are used.
(v) Agents are members of all their assigned coalitions: The DCG algorithm, like the SK and VBFR algorithms before it, guarantee that every coalition distributed to an agent i, includesias a member. In the DCG algorithm, this is achieved by assigning agentito be the first agent in every coalition that it generates (see Definition 79).
This is not the case in any of the DCVC algorithms, as can be seen in Figure 4.1 (that shows the three different types of the DCVC algorithm described in Section2.4.1). This is a weakness for the DCVC algorithm for certain domains. Consider any e-commerce environment where agents (representing a single business) will join with others in a tem- porary coalition if they can gain price discounts and economics of scale. As there maybe anti-monopoly penalties or other considerations, such as communication and logistic costs, then this problem cannot simply be solved by forming the grand coalition. Therefore the agents together will have to calculate the values of the different coalitions to find their most preferable ones. If the variables involved in the calculation of each coalition’s value are public knowledge, then this situation can be treated as a characteristic function game and the coalition value calculation costs can be shared around the agents of the system. In this situation it makes no sense for an agent to calculate the value of a coalitionCthat does not include itself because the agent will gain no benefit ifCwere to form.
When using the DCG algorithm with self-interested agents, possible manipulations could occur by for example, misreporting the value of a certain coalition. In this case the tools of mechanism design [80] may have to be used, where the DCG algorithm can be wrapped
Algorithm Eliminates Communica- tion Eliminates Redundancy Equal Shares Equal Operations Agent in All its Assigned Coalitions SK No No No No Yes
DCVC Yes Yes Yes No No
VBFR Yes Yes No No Yes
DCG Yes Yes Yes Yes Yes
TABLE 4.2: A summary of the comparison between the DCVC, VBFR, SK and DCG algo- rithms that distribute coalition value calculations.
in some form of mechanism that constitutes a Bayesian game. If this mechanism has Bayesian equilibria and the optimal strategy for each agent is to misreport information, then via the revelation principle[76, 119] their exists another payoff-equivalent individ- ually rational mechanism that has an equilibrium where the players truthfully report their types6[76,119].
This discussion, summarised in Table4.2, has shown that the DCG algorithm satisfies all the above properties, which none of the DCVC, VBFR or SK algorithms can do.