Market-Wide Perspective

Contrary to the symmetric case, aggregate deviations between product-specific Shap-ley sharesρ^∗(N,v) and other heuristics do generally not vanish when firms are asym-metric and all customers act against former cartel members. Note, however, that

differences between an allocation based on the Shapley share ρ^∗(N,v) and other heuristics can partially cancel out across products.

In Figure 4.3 we illustrate how ad hoc heuristics and the discretization heuristic fit, i.e., the percentage mis-allocation M^ρ, for different kinds of asymmetries when baseline parameters area=30,γ=2,d=3,n=4,b=d/(3α).

In the two top panels (a) and (b), we allow that firms differ in their market saturation quantity ai. Panel (a) reconsiders the example discussed in Subsection 4.4.3.1. The discretization heuristic ρ^D outperforms ad hoc heuristics for α > 1.4.

With increasing values ofα, only an allocation based on competitive salesρ⁴ comes close to the Shapley based allocationρ^∗(N,v) in both panels. However,ρ^Ddoes much better and achieves smaller mis-allocationM^ρDfor almost all levels of differentiation.

The discontinuous jumps occurring in the product-wise discretization heuristic can be most easily explained by using panel (c) where two firms are twice as efficient as their competitors. When α = 1.55, aggregated overcharge damages are roughly O=321. Then, using a dichotomous approximation requires each firm to contribute share 1/n, that is, H^ρ_i^D = 80.3. Now fix an inefficient firm, say firm 3. Firm 3’s Shapley shares ρ^∗₃(N,v) are (21%; 21%; 35.4%; 20.6%) in the four price overcharges

∆p1,∆p2,∆p3 and∆p4;Φ3=77.2. Thus,ρ^D₃ =25% significantly underestimates firm 3’s own part but slightly overestimates its contribution share when another firm sold the product. Differences between over and underestimation of a firm’s contribution share partially cancel andH^ρ₃^D −Φ3 = 3.1. Hence, M^ρD =4·3.1/321 = 4.3%.¹⁵ Next, considerα = 1.6; then,O = 286. Dichotomous damage shares for a high cost firm abruptly changes to (16.¯6%; 16.¯6%; 25%; 25%) with H^ρ₃^D = 58. Its respective Shapley sharesρ^∗₃(N,v) are (20.8%; 20.8%; 35.9%; 20.4%) withΦ3 =68.7. The product-specific dichotomous approximations now always underestimate firm 3’s shares (except for the case in which firm 4 sold the product), with a huge difference when its home customers sue. The deviation increases toΦ3−H^ρ₃^D =10.7;M^ρD =4·10.7/286=14.9%.

Thus, the sum over the dichotomous approximations that minimize the product-specific deviations – which change continuously in α – only minimizes market-wide deviations by coincidence, not in general. In particular, on a market-market-wide perspective, the “second-best” dichotomous approximation used in our robustness analysis sometimes outperforms the approximation that minimizesP

i∈N |ρ^∗_i −ρ^D_i |. In panels (c) and (d), an allocation based on competitive revenue sharesρ³is very close toΦiand frequently outperforms the discretization heuristic. In panel (c),ρ^D is

15Note that two firms are efficient and two firms inefficient each. When the efficient firms pay too much, the inefficient firms have to pay too little since all discussed allocation rules satisfy efficiency.

(a) ai =aj/3=a (b) a1=a2/3=a3/7=a4/10=a

worst for some values ofα. In this case, however, all heuristics could be used since normalized deviations in panel (c) are all rather small.

When own-price effects of two firms increase by factor η (see panel (e)), some ad hoc heuristics, in particular the one based on competitive revenue shares ρ³, perform well. The discretization heuristic is sometimes best, but can also be worst for a very small range ofη. Nevertheless,ρ^D is close to the Shapley shareρ^∗(N,v) for

all relevant values ofη;M^ρD is consistently below 15%.

Finally, panel (f) involves two firms with increasing cross-price effects. With more intense competition, that is, with increasing values ofβ, only the discretization heuristic is close to the Shapley value in the original game. With rather symmetric firms, an allocation based on profit sharesρ⁵performs relatively good.

To sum up: there is no ad hoc heuristic which is always close to the correct Shapley shares evaluated for (N,v). This differs for a heuristic allocation based on dichotomous damage scenarios. It does not only reflect responsibility, but M^ρD is additionally always significantly below 20% in the discussed cases. Sometimes, it is the only heuristic which is close to the Shapley value of the original game (N,v).¹⁶

4.5 Concluding Remarks

This chapter has argued that a responsibility-based allocation of cartel damages is feasible even without a full-blown merger simulation analysis. This should receive increasing attention in the EU since Directive 2014/104/EU has been transposed into national law throughout the EU.

The question how to economically quantify the relative responsibility of a firm is answered in Chapter 3: ideally use the Shapley value. Ad hoc heuristics could be used to approximate the Shapley share in view of the former’s strong data requirements.

But no heuristic always outperforms the others. Thus, more details on the firms at hand have to be known to argue which heuristic indeed fits best. Then, however, the additional expense to answer the question which coalitions actually caused (or would have caused, as counterfactual market scenarios are evaluated) huge damage, is small. This leads to the proposal of a new heuristic,ρ^D, based on approximation by dichotomous damage scenarios. Heuristicρ^Dhas two main advantages compared to ad hoc heuristics. First, it reflects relative responsibility and second, it rather robustly offers a good approximation of the Shapely value in all considered market scenarios;

from a product but also from a market-wide perspective.

16 Interestingly, although baseline parameters in panels (a) – (f) differ from the ones used in the simulations in Subsection 3.3.4, the ad hoc heuristic which is closest to the Shapley value in the original game stays unchanged. Thus, baseline parameters are rather unimportant to evaluate and select ad hoc heuristics; the asymmetry at hand matters.