How to (Carefully) Breach a Service Contract?

Consider a firm \(\mathcal {S}\) providing support to clients \(\mathcal {A}\) and \(\mathcal {B}\).

The contract \(\mathcal {S} \leftrightarrow \mathcal {A}\) stipulates that \(\mathcal {S}\) must continuously serve \(\mathcal {A}\) and answer its calls immediately. While servicing \(\mathcal {A}\), \(\mathcal {S}\) incurs two costs: personnel fees (salaries) that \(\mathcal {A}\) refunds on a per-call-time basis and technical fees that are not refunded.

The contract \(\mathcal {S} \leftrightarrow \mathcal {B}\) is a pay-per-call agreement where \(\mathcal {S}\) gets paid an amount proportional to \(\mathcal {B}\)’s incoming call’s duration. We consider that the flow of incoming \(\mathcal {B}\) calls is unlimited and regular.

\(\mathcal {S}\) wishes to use his workforce for both tasks, switching from \(\mathcal {A}\) to \(\mathcal {B}\) if necessary. As \(\mathcal {S} \leftrightarrow \mathcal {B}\) generates new benefits and \(\mathcal {S} \leftrightarrow \mathcal {A}\) is the fulfilling of a contracted obligation, \(\mathcal {S}\) would like to devote as little resources as necessary to support \(\mathcal {A}\) and divert as much workforce as possible to serve \(\mathcal {B}\). Hence, \(\mathcal {S}\)’s goal is to minimize his availability to serve \(\mathcal {A}\) without incurring too high penalties.

This paper models \(\mathcal {A}\) as a naïve player. This captures \(\mathcal {A}\)’s needs but not \(\mathcal {A}\)’s game-theoretic interests – which thorough investigation remains an open question.

1. 1.

   i.e. 1–900 call.

2. 2.

   The model can be easily generalized to cases where working for \(\mathcal {B}\) generates a positive margin.

3. 3.

   \(\ell \) represents the legal fees, image negative impact and compensations paid to \(\mathcal {A}\).

4. 4.

   Assume, for instance, that \(x(t)=10\) calls/hour and consider a one second time interval \(\varDelta T = 1/3600\) h. The probability to witness a call during \(\varDelta T\) is indeed \(10 \varDelta T=1/360\). During each one second time interval the probability to witness a call is 1 / 360 and over 3600 s we indeed get an average of 10 calls. Hence, p is indeed the probability to witness a call between time t and \(t+\varDelta T\) when \(\varDelta T\) is very small.

We thank Jean-Sébastien Coron for his contribution to this work.

Authors and Affiliations

1. Paris Center for Law and Economics (CRED), Sorbonne Universités – Université Paris II, 12 Place du Panthéon, 75231, Paris Cedex 05, France

   Céline Chevalier, Damien Gaumont, David Naccache & Rodrigo Portella Do Canto

Corresponding author

Correspondence to David Naccache .

Editors and Affiliations

1. Université du Luxembourg , Luxembourg, Luxembourg

   Peter Y. A. Ryan

2. Ecole normale supérieure , Paris, France

   David Naccache

3. Université Catholique de Louvain , Louvain-la-Neuve, Belgium

   Jean-Jacques Quisquater

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