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Abstract
When a warranty provider outsources warranty servicing to an external service agent this agent may act in a fraudulent manner. In this paper we consider a special case of service agent fraud—with the service agent overbilling the warranty provider for some of the warranty claims. A detailed inspection of a claim may be made to identify whether or not the service agent has committed fraud, but this inspection involves an additional cost to the warranty provider. This cost may be recovered by imposing a penalty on the service agent whenever a fraud is committed and it is detected. This penalty is specified in the maintenance service contract. A game theoretic approach is used to find the optimal overbilling strategy for the service agent and the optimal inspection strategy for the warranty provider. The optimal solution is the mixed strategy Nash equilibrium of a static game between the two parties.
Abbreviations
| BR | = | Best response |
| BW | = | Base warranty |
| DM | = | Decision-maker |
| EW | = | Extended warranty |
| GT | = | Game theory |
| MSC | = | Maintenance service contract |
| NE | = | Nash equilibrium |
| RV | = | Random variable |
| SA | = | Service agent |
| WP | = | Warranty provider |
| List of symbols | ||
| C | = | Cost of each inspection |
| P | = | Penalty incurred by SA if fraud is detected [P > C] |
| p | = | Probability that the SA overcharges a claim resulting in fraud [SA decision variable] |
| q | = | Probability that the WP carries out an inspection [WP decision variable] |
| W | = | Length of warranty |
| N(W) | = | Number of item failures (warranty claims) over the warranty period [RV] |
| λ(t) | = | Failure intensity function |
| Λ(t) | = | Cumulative failure intensity function |
| X | = | Excess fraudulent claim by SA [RV] |
| F(x) | = | Distribution function of X |
| f(x) | = | Density function of X |
| E[X] and E[X2] | = | First and second moments of X |
| Zi | = | Revenue to SA associated with ith warranty claim |
| R | = | Total SA revenue over the warranty period [RV] |
| Yi | = | Revenue to WP associated with ith warranty claim |
| S | = | Total WP revenue over the warranty period [RV] |
| ϕ | = | Risk aversion parameter for SA |
Abbreviations
| BR | = | Best response |
| BW | = | Base warranty |
| DM | = | Decision-maker |
| EW | = | Extended warranty |
| GT | = | Game theory |
| MSC | = | Maintenance service contract |
| NE | = | Nash equilibrium |
| RV | = | Random variable |
| SA | = | Service agent |
| WP | = | Warranty provider |
| List of symbols | ||
| C | = | Cost of each inspection |
| P | = | Penalty incurred by SA if fraud is detected [P > C] |
| p | = | Probability that the SA overcharges a claim resulting in fraud [SA decision variable] |
| q | = | Probability that the WP carries out an inspection [WP decision variable] |
| W | = | Length of warranty |
| N(W) | = | Number of item failures (warranty claims) over the warranty period [RV] |
| λ(t) | = | Failure intensity function |
| Λ(t) | = | Cumulative failure intensity function |
| X | = | Excess fraudulent claim by SA [RV] |
| F(x) | = | Distribution function of X |
| f(x) | = | Density function of X |
| E[X] and E[X2] | = | First and second moments of X |
| Zi | = | Revenue to SA associated with ith warranty claim |
| R | = | Total SA revenue over the warranty period [RV] |
| Yi | = | Revenue to WP associated with ith warranty claim |
| S | = | Total WP revenue over the warranty period [RV] |
| ϕ | = | Risk aversion parameter for SA |
Acknowledgments
We thank the two anonymous reviewers for their constructive comments and suggestions which helped us to improve an earlier version of the paper.