Before presenting our trust-based mechanism (TBM), we first introduce some new no- tation. Let the sum of utilities of all agents in a system given an allocation K and a completion vectorγ be denoted asU(K,θ, γ) =Pi∈Ivi(K, θi, γ)−Pi∈Ici(K, θi). Then the expected utilityU(K,θ, γ) before the allocation is carried out isE[γ|K,ti][U(K,θ, γ)] where θ is the vector containing all agent types. We also denote the marginal con- tribution of the agent i to the system given an efficient allocation Kb∗ as mc
i = U−i(Kb∗,θ, γ)−maxK∈K £ U−i(K,θ−i, γ) ¤ where maxK∈K £ U−i(K,θ−i, γ) ¤ is the overall expected utility of the efficient allocation that would have resulted if agent i were not present in the system. Now, we can detail TBM:
1. Find the efficient allocationKb∗ such that:
b
K∗ = arg max
K∈KU(K,θ, γ) (6.7)
This finds the best allocation; that is, the one that maximises the sum ofexpected utilities of the agents, conditional on the reports of the agents. We note here that we do not take into consideration the reward functions of the agents when calculating the overall utility since these rewards are from one agent to another and therefore do not make a difference when calculating the overall utility of the agents.
2. We now calculate the efficient allocation that would have resulted if an agent i’s report taken out:
K−∗i= arg max
K∈KE[γ|K,ti0][U(K,θ, γ)] (6.8)
, where t0
i =g(ηb\ηbi). This computes how ηbi affects which allocation is deemed
efficient.
3. We now find the effect that an agent’s ηbi has had on its marginal contribution.
Thus, find
Di=U(Kb∗, .)−U(K−∗i, .) (6.9) This distils the effect of an agent’s ηbi reports.
4. GivenK∗, the paymentr
i made to the agentiis then:
ri =mci−Di (6.10) Naturally, if ri is negative it implies that i makes a payment to the centre. The first part of the payment scheme,mci, calculates the effect that an agent’spresence
has had on overall expected utility of the system. We also subtractDi to take into account the effect that an agent’s POS report has on the chosen allocation. This is in line with the intuition behind VCG mechanisms in which an agent’s report affects the allocation but not the payment it receives or gives.
We will now prove each of the properties of TBM in turn whilst intuitively explaining why the mechanism has the aforementioned properties.
6.5.1 Properties of our Trust Based Mechanism
Given the mechanism presented in the previous section, we now prove its main properties in the following order: incentive compatibility, efficiency, and individual rationality.
Proposition 1. TBM is incentive-compatible in ex-ante Nash Equilibrium.
Proof. We first need to calculate the expected utility, E[γ|K,ti][ui(K, θi, γ)], that an agent derives from TBM because the goal of a rational agent is to maximise its expected utility. We note here that we are assuming that the agent is myopic in that it is only concerned with its current expected utility given the cost vector, c(K,θ), the value vector,v(K,θ), and the trust vectort. The expected utility that an agent,ui(Kb∗, θ
i, γ), derives from an efficient allocation, as calculated from equation 6.7, given the reports of all agents in the system is:
ui(Kb∗, θi, γ) =E[γ|Kb∗,t i] £ vi(Kb∗, θi, γ) ¤ −ci(Kb∗, θi) +mci(Kb∗, θi, γ)−Di =E[γ|Kb∗,ti] £ vi(Kb∗, θi, γ)−bvi(Kb∗, θi, γ) ¤ − ³ ci(Kb∗, θi)−bci(Kb∗, θi) ´ + U(K−∗i,θ, γ)−max K∈K £ U−i(K,θ−i, γ)¤ (6.11) ¿From 6.11 we will firstly prove the following lemma:
Lemma 1. An agent has an equilibrium strategy to reveal its observed POS values.
Proof. We consider how ηbi affects ui(Kb∗, θi, γ). From equation 6.11 we observe that
b
ηi cannot affect U(K−i,θ, γ)−maxK∈K
£
U−i(K,θ−i, γ)
¤
. Thus, an agent only has an incentive to lie so thatKb∗ is selected such that:
E[γ|Kb∗,t i] £ vi(Kb∗, θi, γ)−vbi(Kb∗, θi, γ)¤− ³ ci(Kb∗, θi)−bci(Kb∗, θi) ´
is maximised. If an agent reveals its cost and valuation truthfully i.e. vb(.) = v(.) and
b
reporting of ηbi. If however, an agent is to gain from such an untruthful reporting, it needs to set either bv(.) < v(.) and bc(.) > c or both. However, doing so would decrease the chance ofisuccessfully allocating a task or winning an allocation. Therefore,iwould not reveal untruthful values forbc(.) andbv(.). Moreover,iwill actually report truthfully its ηbi since this allows the centre to choose those agents that i deems to have a high
POS (as well as helping other agents choose i as having a perception close to theirs). Thus, reportingηbi =ηi is an ex-ante Nash equilibrium strategy.
Given lemma 1, we can now show that TBM is incentive compatible. Suppose an agent is truthful about bv(.) and bc(.). Then it derives as utility:
U(K−∗i,θ, γ)−max K∈K
£
U−i(K,θ−i, γ)¤
Now assume that the agent lies aboutbv(.) andbc(.) so as to increase its utility. This then means that: E[γ|Kb∗,ti] £ vi(Kb∗, θi, γ)−bvi(Kb∗, θi, γ) ¤ − ³ ci(Kb∗, θi)−bci(Kb∗, θi) ´ +U(K−0 i,θ, γ)> U(K−∗i,θ, γ) (6.12) where K0
−i is the efficient allocation found withbc(.) and bv(.) without the report of ηi.
However, as argued earlier, an agent would not report a lower value or a higher cost. Thus E[γ|K,ti]£vi(Kb∗, θi, γ)−bvi(Kb∗, θi, γ) ¤ − ³ ci(Kb∗, θi)−bci(Kb∗, θi) ´ ≤0 (6.13) Furthermore, by the maximisation of step 2 of TBM:
U(K−0 i,θ, γ)< U(K−∗i,θ, γ) (6.14) if all other agents report truthfully. Thus, TBM is incentive-compatible in a Nash equilibrium.
Proposition 2. TBM is efficient.
Proof. Given that the agents are incentivised to report truthfully (proposition 1), the centre will calculate the efficient allocation according to equation 6.7 (i.e. Kb∗=K∗).
Proposition 3. TBM is individually-rational (in expected utility).
Proof. We need to show that the expected utility of any agent from an efficient allocation K∗ is greater than if the agent were not in the scheme (i.e. u
i(K∗, θi, γ) ≥ 0). As a result of the inherent uncertainty in the completion of tasks, we cannot guarantee that the mechanism will be ex-post individually-rational for an agent. Rather, we prove that the mechanism is individually-rational for an agent if we consider expected utility.
Given truthful reports, the utility of an agent from equation 6.11 is U(K∗ −i,θ, γ)− maxK∈K £ U−i(K,θ−i, γ) ¤
. The first maximisation is carried out without the reports
η−ii ,whereas the second maximisation is carried out over the set of agentsI \i. Thus, the second maximisation is carried out over a smaller set than the first one. As a result:
max K∈K £ U−i(K,θ−i, γ) ¤ ≥U(K−∗i,θ, γ) (6.15) such thatui(K∗, θi, γ)≥0. 6.5.2 Instances of TBM
TBM can be viewed as a generalised version of the VCG mechanism in which there exist uncertainties about whether a set of agents will carry out an allocation and about the relevance of reports of POS by agents. In this section, we demonstrate its generality by analysing two specific instances of the mechanism.
6.5.3 Self-POS Reports Only
The non-combinatorial mechanism developed in (Porter et al., 2002) is a special case of TBM. Specifically, agents only report on their own POS (i.e. ηbi=bηii) and agents assign a relevance of 1 to reports by all other agents. However, since in their model their is no notion of varying perceptions of success, we need to introduce the notion of areport agent that has v(K, .) = 0 and c(K, .) = ∞. This acts as a proxy to agents reporting the ex-post POS to the centre. This also caters for the problem of single POS reports as there is then no measure of tji once j’s report is removed (and hence U(K∗
−i, .) is undefined). The centre then calculates the efficient allocation as:
K∗ = arg max K∈K h U(Kb∗,θ, γ) i (6.16) and the payment to the agent i is ri =mci−Di =mci. The term Di = 0 since, as a result of the report agent,U(Kb∗, .) =U(K∗
−i, .) (becausetis equal in both cases are the same).
6.5.4 Single-Task Scenario
Consider the single task scenario (as presented in table 6.1) where an agent kproposes a single taskτk. Using equation 6.7, the efficient allocation is then simplified to:
K∗ = arg max K∈K
h
U(Kb∗,θ, γ)
The payment to agent i, from equation 6.10, is then: ri=E[γ|K∗ −i,tk] £ vk(K−∗i, θk, γ) ¤ +bci(Kb∗, θi, γ)− P i∈Ibci(K−∗i, θi, γ) −maxK∈K · E[γ|K,tk][vk(K, θk, γ)]− P j∈−ibcj(K,θ−i, γ) ¸ (6.17)
Since the above single-task scenario is an instance of the TBM, it is still incentive compatible. Therefore, when applying the above allocation scheme to the example, we can take the reported values of the agents as being truthful. Given this, the efficient allocation is agent 3 getting to do the task. Then, we need to check whether agent 3’s report has made itself more attractive. To do so, we remove the report of agent 3 and end up with agent 4 having a trust vector ti
4 = [0.5 1.0 0.9] which again
leads to agent 3 being allocated the task. Thus agent 3 will get an expected utility of 210∗0.8667 − 50 + 50 −130−50 = 2. Agent 1 and 2 no longer have an incentive to lie about the POSs since this would not increase their utility. However, suppose that, after the allocation, every type becomes common knowledge. Then agent 2 can deduce that lying about its costs and reported POS would allow its utility to increase. This would have been maximised when agent 2 reports bc2(.) = 110 and ηb32 = 0. However, before
the allocation is carried out and payments are made, agent 2 would not know about the private types of other agents and may reduce its chance of deriving a positive utility by reporting bc2(.) > c2(.). Furthermore, agent 2 does not report ηb2−2 < η2−2 since then
u2(.) = 0 even if it wins the allocation. A similar argument applies to agent 1. Thus, the mechanism has an ex-ante Nash Equilibrium of truthful reporting.