Stable allocations for collaborative choice-based pricing intransport markets
V. Lurkina,∗ , L. Schlicherba University of Lausanne, HEC Lausanne Faculty of Business and Economics, Lausanne,

Switzerland
virginie.lurkin@unil.ch,∗ Corresponding authorb Eindhoven University of Technology, Department of Industrial Engineering and Innovation
Sciences, Eindhoven, The Netherlands
l.p.j.schlicher@tue.nl
Extended abstract submitted for presentation at the 11th Triennial Symposium on
Transportation Analysis conference (TRISTAN XI)
June 19-25, 2022, Mauritius Island

March 31, 2022
Keywords: game theory, choice-based pricing, cooperative game, core, allocation rules

1

Introduction

Horizontal agreements may qualify for exemption if they create sufficient pro-consumer benefits
that outweigh the anti-competitive effects (Article 101(3), TFEU). Besides, they should not
eliminate the competition in the relevant market, implying that participants should have a small
market share and their combined market share should not exceed a specified limit (e.g., 20%
in the Netherlands). A recent example of such an exemption stems from the Webtaxi case in
Luxembourg, where the competition authority allowed various taxi companies to use a pricing
algorithm to determine their taxi prices. While it was acknowledged that the joint use of suchan algorithm constituted a situation of price fixing (i.e, a price agreement between competitors),
it was decided that the agreement could be exempted since they expected huge pro-consumer
benefits, mainly less waiting time, and lower prices, as well as more rides for drivers (Bostoen(2018)). At the same time, the combined market shares of the taxi operators would remain far
below the threshold set by the Luxembourg authorities.
Inspired by this joint profit (re)allocation problem, we study a setting in which a set oftransport operators (e.g., micromobility startups) can collaborate and decide at what price tooffer sustainable urban mobility solutions (e.g., electric scooters or bikes) to a pool of travelers.
To better reflect the decisions of these travelers, we assume that they choose among the services
offered according to a multinomial logit model, one of the most widely-used disaggregate demandmodel (Ben-Akiva & Bierlaire (2003)).
Our work fits within the literature on choice-based pricing models, i.e., studies that integrate customer’s choice behavior within pricing problems. These models are mathematicallycomplex since they are nonlinear and non-convex in prices. Several equilibrium studies havebeen published on choice-based pricing problem involving multiple competitive firms (see e.g.,
Lin & Sibdari (2009), Levin et al. (2009), Morrow & Skerlos (2011), Bortolomiol et al. (2021)).
However, as already mentioned, unlike these studies and inspired by horizontal agreement exemptions, we assume that the firms can collaborate and collectively decide at what price to offer
their services. Cooperative game theory is then the most appropriate methodology to adopt to
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allocate the associated joint profit between the firms. Our paper is therefore enrolled in the lineof this existing literature on cooperative games inspired by real-life settings in transport.
By considering a choice model on the demand side, our setting involves pricing decisions that
better capture the supply-demand interactions between the operators objective of maximizing
their expected revenue and the travelers objective of maximizing the expected utility (Sumida
et al. (2019)). To be in line with the conditions associated with the horizontal agreement exemptions, we assume that the transport operators set their prices in such a way that the total
joint profit is maximized and their total market share remains constant, and as such remains
below the authorized limit. In the next section we present a cooperative game for this setting,the transport choice (TC) game, and introduce various intuitive allocation rules.

2

A Cooperative Transport Choice Game

We consider a setting in which a group of homogeneous travelers is buying mobility services froma set of N ⊆ N transport operators. Each operator i ∈ N offers one micromobility service (e.g.,a e-bike or a segway) against price pi ∈ R+ and cost price ci ∈ R+ . The mobility choices oftravelers are represented using the logit model and we can therefore define the market share oftransport operator i as the share of travelers that opts for mobility service i ∈ N . Following thelogit model, this is given byeαi −βpi
P.
(1)
1 + j∈N eαj −βpj
where αi ∈ R+ is an alternative-specific constant and β ∈ R+ a price sensitivity parameter.
Given this market share, the profit of transport operator i ∈ N is defined by:(pi − ci ) ·eαi −βpi
P.
1 + j∈N eαj −βpj(2)

We summarize this setting by tuple θ = (N, p, c, α, β) with N the set of transport operators,p = (pi )i∈N the vector of prices, c = (ci )i∈N the vector of cost prices, α = (αi )i∈N the vectorof alternative-specific constants, and β the price sensitivity parameter. We refer to θ as thetransport choice situation and let Θ be the set of all possible transport choice situations.

An example Let θ ∈ Θ with N = {1, 2, 3}, p = (6, 8, 15), c = (8, 4, 1), α = (1, 0.5, 1.5) and
β = 0.36. The market shares and associated profits corresponding to the initial prices of thetransport operators are presented in Table 1.
⋄iPrice imarket share i
profit i

1
6.0
0.220
-0.440

2
8.0
0.065
0.260

3
15.0
0.014
0.199

Table 1 – Prices, market shares and profits of the transport operators of situation θ
For each TC situation θ ∈ Θ, we now introduce a cooperative game (N, v θ ), where N represents the set of players (i.e., transport operators) and v θ represents the characteristic valuefunction. In this game, v θ (M ) reflects the joint profit coalition M ⊆ N \{∅} can realize. This
joint profit is obtained by taking into account that (i) the sum of the market shares of the players
in M remains stable (i.e., the new vector of prices should be such that the sum of their market
shares remains the same) and (ii) all players outside coalition M (i.e., players in N \M ) keep
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their initially set prices. So formally, for every TC situation θ ∈ Θ, the associated cooperativetransport choice (TC) game (N, v θ ) is defined byXv θ (M ) = max

x∈RM

s.t.

X
i∈M

(xi − ci )i∈Meαi −βpi
P
1 + j∈N eαj −βpj1+eαi −βxiP
αj −βxj +αj −βpjj∈M e
j∈N \M e

P

!
=

X
i∈M

eαi −βxi
PP
1 + j∈M eαj −βxj + j∈N \M eαj −βpj!

(3)

for all M ⊆ N \{∅} and v θ (∅) = 0.
For every TC situation θ ∈ Θ, we can show that the optimal joint profit of any coalition
M ⊆ N \{∅}, for all M ⊆ N \{∅}, is given by
DM (p)v (M ) =lnβ(DN (p) + 1)θ

DM (c)DM (p)

P
where DM (x) = i∈M eαi −βxi . The proof of this results consists of three steps. First we
relate our original optimization problem to another optimization problem that has a much simpler form of constraint. Then, we identify an optimal price vector and the associated optimalvalue for this new optimization problem, by using a Lagrangian type of optimality result from
Bazaraa et al. (2013). Finally, we relate back these outcomes to our original problem.
In Table 2, you can see the optimal price vector and the corresponding market share andprofit per transport operator for the example above.
i
optimal price imarket share i
profit i1
13.980
0.012
0.074

2
9.980
0.032
0.190

3
6.980
0.255
1.523

Table 2 – Prices, market share and profits of the transport operators of situation θ
From Table 1 and Table 2, we learn that the joint profit, which is 1.787, exceeds the sum of
individual profits without collaboration, namely −0.440 + 0.260 + 0.199 = 0.019. However, atthe same time, we also observe that the individual profit of transport operator 2 decreases (from0.260 to 0.190). So, in case of collaboration among the three transport operators, it would benatural that operator 1 and operator 3 would compensate operator 2 in some way.
The coalitional values of TC game (N, v θ ) corresponding to our example are represented inTable 3 below. You can see that the coalitional values of the individual coalitions (-0.440, 0.260
and 0.199) match with the profits of Table 1, and that the coalitional value of the grand coalition(1.787) matches with the sum of the profits of Table 2.
Mv θ (M )

{∅}
0

{1}
-0.440

{2}
0.260

{3}
0.199

{1,2}
0.230

{1,3}
1.485

{2,3}
0.756

{1,2,3}
1.787

Table 3 – Coalitional values of game (N, v θ )The next section shows how to distribute the overall joint profit amongst the different operators.
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4

Allocation rule

It can be showed that some well-known allocation rules such as proportional rules and the Shapley
value do not always belong to the core. One reason could be that these allocation rules do not
explicitly compensate for the exchange of market share between players. Therefore, we present an
allocation rule that does explicitly compensate for this exchange of market share. The allocation
rule first allocates to each player the profit he/she generates under full collaboration, i.e., player
{i}

∗

)
i ∈ N receives (p∗i − ci ) DDN (p(p∗ )+1 . Thereafter, we identify for each player i ∈ N the increase (or {i} ∗
)
D{i} (p)decrease) in the market share, which is DDN (p(p∗ )+1 − D N (p)+1 . Player i then receives a price ϕfor each exchanged unit of market share, and pays the same price for each extra unit of market
share. Formally, for each θ ∈ Θ and associated TC game, the market share exchange (MSE) ruleis given by!
{i} (p∗ ){i} (p∗ ){i} (p)D
D
D,
M SEi = (p∗i − ci ) N ∗−ϕ−
D (p ) + 1
DN (p∗ ) + 1 DN (p) + 1

where the price ϕ is given byv θ (N ) − v θ (N )
 N
 ,
b

ϕ=

D (p)
DN (p)+1

with θb = (N, p, (pi − 1/β)i∈N , α, β), i.e., θb is a TC situation with a constant marginal profit
for all operators (1/β) and with the same total market share as θ (D(p)/(D(p) + 1)). Players
cannot gain from such a TC situation (because pi −ci = β1 for all i, j ∈ N ). Hence, the numeratorof ϕ represents the total additional return that is gained compared to a TC situation with thesame total market share and where collaborating is not beneficial at all. This total additional
return is then divided by the total market share. Indeed, ϕ can be recognized as the additional
return per unit of market share. It can be showed that the allocations of the MSE rule always
belong to the core. The proof of consists of two steps. First, we can show that the MSE satisfies
efficiency, which follows by construction of MSE. Thereafter, we can show that MSE satisfies
stability.
The allocations of the proportional allocation rules for (N, v θ ) are reported in Table 4.
⋄
P
M
{1}
{2}
{3}
{1, 2}{1, 3}{2, 3}{1, 2, 3}v θ (M )-0.440
0.260
0.199
0.230
1.485
0.756
1.787

i∈M xi

I-PROP-42.10124.85919.029-17.242-23.072
43.889
1.787

M-PROP
1.314
0.388
0.085
1.702
1.399
0.473
1.787

Table 4 – Illustration of proportional allocation rules
I-PROP is not in the core, since I-PROP1 + I-PROP2 = −17.242 < 0.230 = v θ ({1, 2}). Thatmeans, players 1 and 2 together can earn more by breaking up and forming a new coalition
together. Similarly, M-PROP is not in the core, since M-PROP3 = 0.085 < 0.199 = v θ ({3}).
That means, player 3 is better of by working individually. It can also be showed that the Shapley
value, SV = (0.407, 0.392, 0.989), does not belong to the core.
Unlike these rules, the allocation of the MSE rule that is given by M SE = (0.738, 0.296, 0.753)
does belong to the core.
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