0, if q < qminb ,
v(q), if qbmin ≤ q ≤ qbmax, v(qbmax), if q > qmaxb ,
where v(q) is the consumers’ common value function and qbmin and qbmax are con-sumer specific parameters of that function. Here we assume simply that v(q) = q.
This means that each consumer has a minimum useful quality qminb and any ser-vice that does not offer at least this is worthless (Vb = 0). On the other hand, the consumer also has a maximum useful quality, qbmax, which gives him his full utility. Any improvement above this level does not increase the value of the service to the consumer in question. The parameters qbmin and qbmax are selected for each consumer independently at random from U nif orm(0, 0.5) and U nif orm(0.5, 1.0) respectively. The consumer’s reservation price for a given service is then equal to its value.
3.3 Decommitment
Since the original contract is always beneficial to both parties (Us > 0 and Ub > 0) they would not consider abandoning it without some external force. We therefore
introduce a possibility of an adverse impact that decreases the value of the contract to the party in question. This decrease may make the contract counter-productive to the affected party or parties and he or they may want to decommit. We use ab and as to denote the probability that the buyer or seller (respectively) will be affected.
For the provider, the decrease means that the cost of providing the service increases by amount Ls and this will decrease its utility by the same amount. He will then need to make a decision on whether or not to decommit from the contract in this new situation. The decision is influenced by the decommitment fee fs. We assume that the provider s will decommit at turn t if and only if:
Us(contract|Ls = l) < Us(tdecommit = t) p − cs− l < −fs− Cs(qs, t).
where Us(tdecommit = t) is the seller’s utility, when he decommits at turn t and l is the amount the utility decreases. Here we use the following ten values l ∈ {0.1, 0.2, ..., 1.0}. So, the seller decommits if the decreased utility is lower than the cost it has already paid and the decommitment fee it has to pay to get out of the contract. The seller learns of the loss at some point tl (selected at random) between the time the contract was formed tcontract and the time it was due to be performed (tdelivery) excluding both of the extremes. However, we assume that this loss itself is always avoidable, if the contract is abandoned before it is delivered.
This means that the additional cost has to be paid just before the delivery. It is not possible that this additional cost is paid if there is no delivery.2 There can only be one effect per party and the effect is always final. All possible moments for learning of the effect are equally likely, i.e. tl ∼ U nif orm(tcontract+ 1, tdelivery− 1).
If tcontract+ 1 > tdelivery− 1 we assume that there can be no loss.
The same applies to the buyer, except for two important differences. For the buyer, the impact l decreases the value of the contract (Vb(q)). The fact that the effect can be avoided also here by decommitting at any time before the deadline is clear. However, because the buyer can, in many types of service, just ignore the service delivered, the value cannot be enormously negative. We therefore assume that the impacted value cannot be lower than −0.05 (Vb(q) ≥ −0.05). This small negative value would then come from accepting the service and disposing of the results. This means that the utility of the buyer can never go below −0.05 − p. We
2However, it is possible that the seller has to pay the original cost even without a delivery as explained earlier. Only the extra cost is tied to the actual performance and will always be avoided, if there is no performance.
do not make a similar assumption with the seller, because the cost of producing the service can (in theory) increase without any limit (hardware failures, resource shortages and strikes can make the service very expensive to perform). The second difference is that for the buyer, always Cb = 0 (for the seller Cs≥ 0).
Since, in this chapter, we are interested in the system-level performance and com-mon good, we use the sum of utilities of all buyers and sellers in the market as a performance measure. We chose the sum of utilities because it is the simplest way to measure common good. In addition, it is also the measure used in the law and economics literature.3
The total utility for a buyer and seller pair in the case the seller decommits from the contract at turn t is:
Ub+s(ts decommits = t)
= Ub(ts decommits= t) + Us(ts decommits= t)
= fs+ (−fs− Cs(qs, t)) = −Cs(qs, t).
Similarly, the case where the buyer decommits Ub+s(tb decommits= t) = −Cs(qs, t).
In case both parties decide to decommit at the same turn, we assume that fs = fb = 0. It is clear that the total utility is equal to Cs(qs, t) here as well. The affected parties avoid the utility decrease of the contract, because there is no contract any more, but the decommitter will have to pay the fee (fs or fb) to the victim.4
If the parties decide to perform the contract despite the utility decreases, the total utility is:
Ub+s(contract|Ls = ls&Lb = lb)
= Ub(contract|Lb = lb) + Us(contract|Ls= ls)
= (Vb(qs) − p − lb) + (p − cs− ls) = Vb(qs) − cs− lb− ls.
A decommitment policy is a set of rules that specifies the amount the decommitter (the party decommitting) should pay to the victim (the decommitter’s opponent) in case of decommitment. We will discuss several decommitment policies in this
3In our earlier work (Ponka and Jennings 2007), we used an expected utility of all contracts as a performance measure. However, in the more complicated situations we investigate in this thesis, calculating the expected utility would be much more difficult and we have instead selected the time for adverse impact (potential decommitment time) at random.
4Note that the fee does not change the total utility, just the distribution of wealth in the society. However, as explained earlier, the fee can affect when and if contracts are decommitted from and that can have an impact on the welfare of the society.
work, some of which are only to be used in a certain settings for a certain reason, but we will start by discussing shortly some basic policies that have been used in the literature (or can be easily derived from such policies). The common factor for all these policies is that they are not environment- or issue-specific. These policies are simple and usually they are used when an option of decommitment is needed, but the role of the decommitment policies and fees are not considered in detail.
Specifically, we consider the following:
• Not Allowed : The contracts are absolutely binding and decommitment is not possible.
• Constant: The decommitment penalty f is constant; here we investigate cases where f ∈ {0.00, 0.25, ..., 1.00}.
• Increasing: The decommitment starts with min at tminand increases linearly to max at time tmax. We investigate cases where min = {0.00, 0.25, 0.50}
and max = {0.25, 0.50, ..., 1.00, 1.50, 2.00, 2.50} and min < max. There are three variations (all with tmax = tdelivery):
– Contract Time Only: tmin = t0 and t = tcontract.
– Decommitment Time Only: tmin = t0, and t = tdecommit. – Both: tmin = tcontract and t = tdecommit.
• Constant Price (Andersson and Sandholm 2001): The decommitment fee is a fraction of the price (p). Here we investigate cases where f = {0.5p, 1.0p, ..., 2.5p}.
• Increasing Price: This has the same variations as the increasing policy (con-tract time only and decommitment time only variations were used in (An-dersson and Sandholm 2001)), but the minimum and maximum are fractions of the contract price. We investigate cases where min = {0, 0.25p, 0.5p} and max = {0.5p, 1.0p, ..., 2.5p} and in all cases min < max.
In total, this means that there are 107 variations. Nevertheless, there is still a large number of possible policies and an infinite number of parameter values that we do not investigate. We have tried to choose a reasonable sized selection of the most obvious policies that have been used in literature or that are quite straight-forward variations of those policies. All of these policies are non-compensatory, which means that they do not try to compensate for the losses of the opponent.
We will also introduce two simple compensatory policies that explicitly try to compensate for the losses of the victim, namely:
• Expectation Damages: The fee is the opponent’s expected profit (his utility if the contract is performed properly) plus his costs at decommitment time.
• Reliance Damages: The fee is equal to the opponent’s costs at decommitment time.
We will also discuss some situation-specific variations of these policies. These basic policies rely on complete information about the opponent’s losses but in some cases, we will also discuss some other policies that try to mimic the effect these policies have, but under incomplete information. We will now turn our attention to the specific settings and decisions.