Stackelberg equilibrium premium strategies for push-pull competition in a non-life insurance market with product differentiation

Asmussen, Søren; Christensen, Bent Jesper; Thøgersen, Julie Article Stackelberg equilibrium premium strategies for push-pull competition in a non-life insurance market with product differentiation Risks Provided in Cooperation with: MDPI – Multidisciplinary Digital Publishing Institute, Basel Suggested Citation: Asmussen, Søren; Christensen, Bent Jesper; Thøgersen, Julie (2019) : Stackelberg equilibrium premium strategies for push-pull competition in a non-life insurance market with product differentiation, Risks, ISSN 2227-9091, MDPI, Basel, Vol. 7, Iss. 2, pp. 1-20, https://doi.org/10.3390/risks7020049 This Version is available at: https://hdl.handle.net/10419/257887 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by/4.0/ risks Article Stackelberg Equilibrium Premium Strategies for Push-Pull Competition in a Non-Life Insurance Market with Product Differentiation Søren Asmussen 1,∗, Bent Jesper Christensen 2and Julie Thøgersen 1 1Department of Mathematics, Aarhus University, 8000 Aarhus, Denmark; [email protected].dk 2Department of Economics and Business Economics and the Dale T. Mortensen Centre, Aarhus University, 8210 Aarhus, Denmark; [email protected] *Correspondence: [email protected] Received: 16 January 2019; Accepted: 14 April 2019; Published: 1 May 2019 Abstract: Two insurance companies I1 , I2 with reserves R1(t) , R2(t) compete for customers, such that in a suitable differential game the smaller company I2 with R2( 0 )<R1( 0 ) aims at minimizing R1(t)−R2(t) by using the premium p2 as control and the larger I1 at maximizing by using p1 . Deductibles K1 , K2 are fixed but may be different. If K1>K2 and I2 is the leader choosing its premium first, conditions for Stackelberg equilibrium are established. For gamma-distributed rates of claim arrivals, explicit equilibrium premiums are obtained, and shown to depend on the running reserve difference. The analysis is based on the diffusion approximation to a standard Cramér-Lundberg risk process extended to allow investment in a risk-free asset. Keywords: Stochastic differential game; Product differentiation; Adverse selection; Stackelberg equilibrium 1. Introduction Insurance premiums are typically calculated based on the expected loss, with an added loading depending on distributional properties of the risk (the expected value principle, variance principle, utility premium, etc.). An alternative to these static premium principles is to consider the premium as a dynamic control variable of the insurance company, as suggested in Asmussen et al. (2013) and Thøgersen (2016). In this approach, the individual customer’s problem of deciding whether or not to insure at any given premium offered is modelled explicitly, and the premium is chosen optimally by the insurance company, balancing the resulting portfolio size against revenue per customer in order to minimize ruin probability. The analysis is based on the diffusion approximation to a standard Cramér-Lundberg risk process, extended to allow investment in a risk-free asset. In Asmussen et al. (2019), this idea is extended to a situation where insurance companies compete against each other, and Nash equilibria in premium controls of the resulting stochastic differential game are determined under suitable conditions. However, in some cases, no Nash equilibrium exists. In the present paper, we present a parallel to this analysis dealing with product differentiation, with insurance companies offering different deductibles, and accounting for the possibility of Stackelberg equilibria. Two insurance companies compete against each other such that one company is the leader, choosing its premium first, and the other company is the follower, choosing its premium in response to the leader’s. The setting is slightly modified relative to that in Asmussen et al. (2019), in that we do not consider search and switching costs when modelling the customer’s choice between insurance products. Our main contributions are, first, to establish the existence of Stackelberg equilibrium under suitable conditions on this strategic game between insurance companies, and to identify the restrictions under which this reduces to the special case of Nash equilibrium. To our best Risks 2019,7, 49; doi:10.3390/risks7020049 www.mdpi.com/journal/risks Risks 2019,7, 49 2 of 23 knowledge, this adds at least the following new features to the literature on game theory in insurance: an example of Stackelberg equilibrium in premium controls; a finding of dependence of optimal premiums on reserves; and an occurrence of the phenomenon of adverse selection in a stochastic differential game between insurance companies, i.e., a lower premium charged increases portfolio size but leaves the average customer riskier to the company. In the literature following Taylor (1986), the individual insurance company is frequently modelled as setting its premium in response to the aggregate insurance market, without explicitly considering the analogous behavior of the other companies constituting this market and the resulting strategic interactions. Examples include Taylor (1987) on marginal expense rates, Emms and Haberman (2005) generalizing the deterministic discrete-time analysis of Taylor to a stochastic continuous-time model, Pantelous and Passalidou (2013,2015) using stochastic demand functions in discrete time, and Emms (2007) and Emms et al. (2007), adopting stochastic processes for the market average premium and demand conditions in continuous time. Pantelous and Passalidou (2017) recently found the optimal premium to depend on the company’s reserve in a competitive environment in the sense of this literature, but, again, this is not explicitly a game-theoretic equilibrium in the sense of Nash or Stackelberg, which is where we obtain dependence on reserves. Game-theoretic aspects arise if the other insurers in the market in fact do react to the policy of the individual insurer, with the latter explicitly taking this into account in setting its policy. Market reaction to the individual insurer’s premium is considered by Emms (2011). Explicit games between insurance companies have been studied using non-cooperative game theory, where Cournot games involve volume controls, see, e.g., Powers et al. (1998), whereas premium controls correspond to Bertrand games, e.g., the one-period games in Polborn (1998) and Dutang et al. (2013), who note that one aspect missing in their analysis is adverse selection among policyholders—our analysis includes this. Emms (2012) and Boonen et al. (2018) do consider continuous-time differential games in premium controls, but again based on Taylor (1986) type demand functions of own and market average premium. Boonen et al. (2018) in addition present a continuous-time extension of a one-period aggregate game of Wu and Pantelous (2017), involving a price elasticity of demand or market power parameter, and the individual insurer’s payoff depending on own premium and an aggregate of market premiums. The models are deterministic and open-loop Nash equilibria are determined. In contrast, rather than assuming demand functions, we model the customer’s choice of where to insure directly and find closed-loop or feedback Nash and Stackelberg equilibria in the resulting continuous-time strategic stochastic differential game between insurance companies. The roles of product differentiation via deductibles, adverse selection, and separating equilibrium in our solution are reminiscent of Rothschild and Stiglitz (1976), one of the first applications of game theory to competition in insurance premiums. Besides competition in premiums, game theory has found several other applications in insurance, starting with Borch (1962) on risk transfer. Zeng (2010), Taksar and Zeng (2011), and Jin et al. (2013) consider Nash equilibria of stochastic differential games between insurance companies in reinsurance strategies. The analysis has been extended to non-zero sum games and additional investment controls by Bensoussan et al. (2014), nonlinear risk processes by Meng et al. (2015), ambiguity-aversion by Pun and Wong (2016), and insurance companies with different levels of trust in information by Yan et al. (2017). Stackelberg-type equilibria of stochastic differential games have been studied in Lin et al. (2012), where an insurance company selects an investment strategy while the market (or nature) selects a worst-case probability scenario, and in Chen and Shen (2018), where the game is between insurer and reinsurer, but not as here in a game between insurance companies. For some more remote references, see Asmussen et al. (2019). Stackelberg games were introduced by von Stackelberg (1934), and the theory of stochastic differential Stackelberg games is considered by Yong (2002), Bensoussan et al. (2015), and Shi et al. (2016). Premium competition between insurance companies is likely to arise because the premium charged may affect both portfolio size and revenue per customer. Without market frictions or product Risks 2019,7, 49 3 of 23 differentiation, it might be expected that all customers would simply insure at the company offering the lowest premium. However, this may not be the case in the presence of market frictions. Thus, when choosing which insurance company to contact, customers may face different costs of search and switching, transportation, or information acquisition, or they may simply exhibit differences in preferences. Search frictions have been studied in economics by Diamond (1982), Mortensen (1982), Mortensen and Pissarides (1994), and others. Brown and Goolsbee (2002) studied the effect of internet search on life insurance premiums in US data. Information frictions have been modelled as differences in the cost of obtaining information, e.g., by Salop and Stiglitz (1977). In Asmussen et al. (2019) we study premium competition between insurance companies in the presence of market frictions. In the present paper, we consider instead product differentiation, and for simplicity abstract from market frictions. With the leading example of car insurance in mind, product differentiation may come in several forms. Here, we focus on different deductibles. Other possibilities would be bonus-malus systems, see Denuit et al. (2007), or proportional compensation in deductibles, similar to reinsurance arrangements, see Albrecher et al. (2017). We consider the case of two insurance companies, referred to as I1 and I2 . We allow for product differentiation by letting Ii offer an insurance contract with fixed deductible Ki for a premium pi , i= 1, 2. The deductible measures the quality of the insurance product, so the company offering the lower deductible will be able to charge a higher premium. We assume that there is a financial market consisting of a single risk-free asset with dynamics d Bt=rBt d t , where r is the risk-free interest rate. All excess wealth of customers and reserve of insurers is invested in this asset. There are N customers in the insurance market. We assume that all customers must insure at either I1 or I2 and focus the analysis on the choice between the two companies. This involves several characteristics of both customer and insurance product. We pay special attention to product differentiation and customer risk. The characteristics of an individual customer are unknown to the insurance companies, but their probability distribution known. Based on this distribution, the companies can determine the expected portfolio sizes ni(p1 , p2) and average claim frequencies αi(p1 , p2) in their portfolios as functions of the premiums offered. The gross premium rate of Ii is then ci(p1 , p2) = ni(p1 , p2)pi , and the aggregate claim frequency is λi(p1,p2) = ni(p1,p2)αi(p1,p2). Let r0,i be the initial reserve of company i . For given premiums (p1 , p2) , the reserve of Ii is governed by the dynamics dRi(t) = (µi(p1,p2) + rRi(t)) dt+σi(p1,p2)dWi(t), (1) where (W1,t)t≥0and (W2,t)t≥0are independent Wiener processes, and µi(p1,p2) = ci(p1,p2)−λi(p1,p2)E[(Z−Ki)+] = ni(p1,p2)pi−αi(p1,p2)E[(Z−Ki)+], σ2 i(p1,p2) = λi(p1,p2)E[(Z−Ki)+2] = ni(p1,p2)αi(p1,p2)E[(Z−Ki)+2]. The random variable Z represents claim sizes, assumed to be independent and identically distributed. Thus, (1) can be considered as a diffusion approximation to the Cramér-Lundberg process extended to the case where the insurance companies have access to investment in a risk-free asset. Such diffusion approximations have been used widely, based on the arguments of Iglehart (1969). The aim is to derive value functions for the insurance companies, and determine game-theoretic equilibria. We consider here what we call push-pull competition. We assume that the largest company in terms of initial capital, I1 , selects its premium to try to push the small company away, while the small company tries to pull closer to the large company. For K1>K2 and I2 the leader choosing its premium first, we derive conditions for a Stackelberg equilibrium. The stronger feature of a Nash equilibrium may also occur, and we give conditions for that, but our numerical examples indicate that Stackelberg is the more typical case. Subsequently, for completeness, we briefly sketch the solution in the opposite Risks 2019,7, 49 4 of 23 case, K1<K2 . The claim frequencies of individual customers are considered random to the insurance company, and we obtain explicit solutions for equilibrium premiums in the case of gamma-distributed claim frequencies. The structure of the paper is as follows. In Section 2we analyze the customer’s problem. We proceed to portfolio characteristics in Section 3. In Section 4, we use the portfolio characteristics to find the strategies of I1 and I2 . In Section 5, we obtain explicit solutions in the case of gamma-distributed claim frequencies, and provide numerical examples. Section 6concludes. Some calculations and proofs are deferred to Appendix A. 2. Customer’s Problem The customer has access to the risk-free asset paying interest at rate r . This is the customer’s only source of income, and he/she invests all his/her wealth in this. The customer is exposed to a risk (At)t≥0 , modelled as a compound Poisson process At=∑M(t) n=1Zn , where (M(t))t≥0 is a Poisson process with claim frequency α , and (Zn)n∈N are the claim sizes, assumed to be independent of (M(t))t≥0 . The customer will then reduce this risk by buying insurance. If the customer insures at Ii , then he/she will continuously pay the premium pi , and in return have the claim sizes reduced to at most Ki. The wealth of the customer (wi,t)t≥0when insuring at company ithus has dynamics dwi,t= (rwi,t−pi)dt−dAi,t,wi,0 =w0, where (Ai,t)t≥0 is the compound Poisson process Ai,t=∑M(t) n=1min{Zn , Ki} , and w0 the customer’s initial wealth. We here use similar evaluation criteria and subsequent arguments as in Thøgersen (2016), which we refer to for a more exhaustive treatment. The first step is to realize that the expected present discounted wealth when insuring at Iican be evaluated as Vi=EZ∞ 0exp(−dt)dwi,t=rw0−pi d−r−α d−rE[min{Zn,Ki}], where d>r is a subjective discount rate. If the customer were risk-neutral, he/she would simply choose the insurance company generating maximum expected present discounted wealth. Thus, he/she would prefer Iiover Ijif pi−pj<−αE[min{Zn,Ki}]−E[min{Zn,Kj}]. (2) However, an existence criterion for the insurance industry is that customers are risk averse, and this requires modification of (2) . If Ki6=Kj , the customer will be facing an excess claim size risk when insuring at the company with the higher deductible. Let this additional (or reduced) risk be denoted ze i,j=E[min{Zn , Ki}−min{Zn , Kj}] when insuring at Ii rather than Ij . Please note that ze i,j corresponds to the last factor in (2) , and is positive if Ki>Kj , and vice versa. Let β denote the risk aversion of the customer. By standard arguments of insurance, due to the risk aversion, the customer will be willing to pay a fee to avoid the additional risk. We will take this into account by introducing a personal safety loading ω(β) that the customer is willing to pay to avoid the excess risk present when K16=K2 . This is incorporated in (2) by multiplying the excess risk by ( 1 +ω(β)) . The more risk averse the customer, the higher the safety loading, i.e., ω is non-negative and increasing in β , with ω( 0 ) = 0. Thus, including risk aversion, the customer will prefer I1over I2if p1−p2<−(1+ω(β))αze 1,2 , (3) and conversely, I2over I1if p2−p1<+(1+ω(β))αze 1,2 . (4) Risks 2019,7, 49 5 of 23 In the next section, we use these relations to evaluate the portfolio sizes and average claim frequencies of the respective companies. We remark, however, at this place that in Asmussen et al. (2019) we have presented an in part more sophisticated approach to the customer’s problem involving a finite decision horizon with varying interpretations, but for the sake of simplicity, we have not pursued this aspect here. 3. Portfolio Characteristics The claim frequencies α of the customers will be considered as random to the firm and denoted by Afor a given customer. The case (3) then corresponds to the event Ω=p1−p2<−(1+ω(β))Aze 1,2(5) and (4) to the complementary event Ωc. For I1 the expected portfolio size n1(p1 , p2) and the average claim frequency α1(p1 , p2) take the form n1(p1,p2) = NP(Ω),α1(p1,p2) = E[A|Ω], where Nis the market size. Vice versa for I2, where n2(p1,p2) = NP(Ωc),α2(p1,p2) = E[A|Ωc]. Letting y=p2−p1 (1+ω(β))ze 1,2 , (6) the probability of (5) can for ze 1,2 >0 (corresponding to K1>K2) be evaluated as P(Ω) = P(A<y), so, the portfolio sizes are n1(p1,p2) = NP(A<y),n2(p1,p2) = N(1−P(A<y)). (7) The average claim frequency for I1 is the conditional expected value of the random claim frequency Agiven that the customer insures at I1, i.e., α1(p1,p2) = E[A|A<y], (8) and likewise, for I2, α2(p1,p2) = E[A|A≥y], (9) if y> 0. Otherwise, α1(p1 , p2) = 0 and α2(p1 , p2) = E[A] if y< 0. The criterion y> 0 for obtaining information from the customers’ choices stems from the assumption ze 1,2 > 0, which indicates that I2 offers a better product than I1 , and therefore the premium p1 should not exceed p2 . Otherwise, every customer would obviously choose to insure at I2. In case ze 1,2 < 0, which means that I1 offers a better insurance product, i.e., a lower deductible, K1<K2, then by symmetry n1(p1,p2) = NP(A≥y),α1(p1,p2) = E[A|A≥y], n2(p1,p2) = NP(A<y),α2(p1,p2) = E[A|A<y], if y> 0. Otherwise, if y< 0, then I1 would offer a lower premium for a better product, and would hence win the entire market of customers. Risks 2019,7, 49 6 of 23 4. The Strategies of the insurance Companies—Push and Pull We now consider the optimization problems of the insurance companies. A control π= (π1 , π2) is a set π1(t) , π2(t) of premium strategies where π1(t) , π2(t) denote the premiums set by the companies at time t . As in much of stochastic control theory, we will only consider Markovian (also called feedback) strategies π , meaning that π1(t) , π2(t) only depend on the current value δ of the difference ∆π(t) = Rπ 1(t)−Rπ 2(t) between the corresponding controlled reserve processes Rπ 1 , Rπ 2 . That is, we can write π1(t) = pπ 1(∆π(t)) , π2(t) = pπ 2(∆π(t)) for suitable functions pπ 1 , pπ 2 . Since the (uncontrolled) reserves have the dynamics (1), this makes ∆π(t)a diffusion process, d∆π(t) = µπ(∆π(t)) dt+σπ(∆π(t)) dW(t), (10) where µπ(δ) = µ1pπ 1(δ),pπ 2(δ)−µ2pπ 1(δ),pπ 2(δ)+rδ, σπ(δ)2=σ1pπ 1(δ),pπ 2(δ)2+σ2pπ 1(δ),pπ 2(δ)2, and W= (W1−W2)/√2 is again a Wiener process. Without loss of generality, we take ∆(0) = ∆π(0) = r0,1 −r0,2 >0, i.e., I1 is the large company and I2 the small. The large company seeks to maximize the reserve difference (to push the competitor further away), while the small company seeks to minimize the same (to pull closer to the competitor), each taking the current reserve difference as the state variable. The optimality criterion is to consider a fixed interval [`d , `u] with `d<∆(0)< `uand let τ(π) = inft>0 : ∆π(t)6∈ [`d,`u],Vπ(δ) = Pπ∆π(τ(π)) = `u∆(0) = δ. Then the large company I1 chooses π1 to maximize the probability Vπ(∆( 0 )) to exit at the upper boundary, and the small I2 chooses π2 to minimize Vπ(∆( 0 )) , or equivalently to maximize the probability 1 −Vπ(∆(0)) to exit at the lower boundary. Remark 1. The feedback assumption implies that this is equivalent to maximizing (minimizing) Vπ(δ) for all `d<δ< `u. Given that deductibles are different, one of the firms offers a product of higher quality (lower deductible) than the other. Therefore, the sequence of the game matters, and so a Stackelberg game is considered, where the companies compete sequentially. The sequence of the game is that at any time t 1. The insurance company with the better product (i.e., lower deductible) is the leader and thus plays first. 2. The insurance company with the lower quality product is the follower, and plays second, instantly after observing the leader’s choice. If I2 (the smallest firm) is the leader and I1 the follower (i.e., K1>K2 ), then a Stackelberg equilibrium is defined as a strategy pair (π∗ 1,π∗ 2)satisfying π∗ 1=b π1(π∗ 2)and V(π∗ 1,π∗ 2)≤V(b π1(π2),π2)for all π2, (11) where b π1(π2) = arg supπ1V(π1,π2) . This case, K1>K2 , is relevant when the company offering the lower deductible is not able to attract sufficiently many high-risk customers (who need this extra protection) to become the largest company. We briefly discuss the opposite case below, in Remark 6. The Stackelberg equilibrium concept involves backward induction. First, the optimal response of the follower is determined as a reaction function. Next, the leader inserts the reaction function of the follower into its optimization problem, and solves for the best first move. As the game evolves in Risks 2019,7, 49 7 of 23 continuous time, the reserve difference changes. At each instant, each firm reconsiders its strategy, taking the running reserve difference as the state variable, and taking into account the future strategies of both companies, as long as the reserve difference remains in [`d , `u] . The criteria for a Stackelberg equilibrium are less strict than the ones for the more common Nash equilibrium, defined as a strategy pair (π∗ 1,π∗ 2)satisfying V(π∗ 1,π∗ 2)≥V(π1,π∗ 2)for all π1and V(π∗ 1,π∗ 2)≤V(π∗ 1,π2)for all π2, (12) i.e., neither firm has an incentive to deviate from its strategy unilaterally. We later specify the specific (second order) criteria for our solution for both types of equilibrium. We next quote from Asmussen et al. (2019) some results that will allow replacing optimization problems in the space of functions p1 , p2 by the more elementary problem of pointwise maximization/minimization of the real-valued ratio κπ(δ) = µπ(δ) σπ(δ)2(13) between the drift and variance of the reserve difference process in (10). Lemma 1. Let µ(x) , σ2(x) be bounded and measurable functions on an interval (`d , `u) such that inf`d<x<`uσ2(x)> 0and let X , W be defined on a suitable probability space such that W is a standard Brownian motion and X(t) = δ+Zt 0µX(s)ds+Zt 0σX(s)dW(s)(14) for some δ∈(`d,`u). Define further κ(x) = µ(x)/σ2(x), s(y) = expn−2Zy `d κ(z)dzo,S(δ) = Zδ `d s(y)dy and τ=inft:X(t)6∈ (`d,`u). Then: (i) PX(τ) = `u=S(δ)/S(`u). (ii) For a given function κ on [`d , `u] and a given δ∈[`d , `u] , let ϕ(κ) denote the r.h.s. in (i). Then κ0≤κ1 implies ϕ(κ0)≤ϕ(κ1). By slight abuse of notation, define κ(p1,p2;δ) = µ1(p1,p2)−µ2(p1,p2) + rδ σ2 1(p1,p2) + σ2 2(p1,p2),p1,p2≥0, `d≤δ≤`u. (15) To ease notation here and in the following subsections, we use the notation κ0 i(p1,p2;δ) = ∂ ∂pi κ(p1,p2;δ),κ00 ij(p1,p2;δ) = ∂2 ∂pi∂pj κ(p1,p2;δ), for partial derivatives, where the number of primes indicates the number of times the function is differentiated, and the subscript specifies with respect to which variable. 1 1 The standard notation avoids the primes and considers the subscript as sufficient, but we want to emphasize the differentiation here in order to avoid confusion with other notation in the paper, e.g., µ1(p1,p2)and µ2(p1,p2). Risks 2019,7, 49 8 of 23 Now consider the resulting drift and variance of the reserves in (1), focusing on the case K1>K2 . Writing zi=E[(Z−Ki)+] and z2 i=E[(Z−Ki)21{Z≥Ki}] , it follows from (1) and Section 3that the drift and variance for the reserve of I1can be written as µ1(p1,p2) = NP(A<y)p1−E[A|A<y]z1, σ2 1(p1,p2) = NP(A<y)E[A|A<y]z2 1, and for I2, µ2(p1,p2) = NP(A≥y)p2−E[A|A≥y]z2, σ2 2(p1,p2) = NP(A≥y)E[A|A≥y]z2 2, with y given by (6) . These expressions show that the denominator σ2 1(p1 , p2) + σ2 2(p1 , p2) in (15) depends on the controls p1 , p2 because so does y and K16=K2 implies z2 16=z2 2 (if K1=K2=K then σ2 1(p1 , p2) + σ2 2(p1 , p2) reduces to NE[A]E[(Z−K)+] ). Therefore, we need to optimize over the entire κfunction (15) and not just the difference in drifts νas in Asmussen et al. (2019). From (6) , by lowering the premium p1 , I1 (with a high deductible in their product) can increase y and thereby portfolio size n1(p1 , p2) , for given p2 , but at the expense of simultaneously increasing average claim rate α1(p1 , p2) , leaving the combined effect on the drift µ1(p1 , p2) in (1) of sign that may go either way in general. Thus, there is a tradeoff, reflecting the adverse selection problem, cf. Rothschild and Stiglitz (1976), i.e., lowering the premium brings more but riskier customers. In contrast, by lowering its premium p2 for given p1 , I2 (offering the lower deductible) can lower y and thereby simultaneously increase portfolio size n2(p1 , p2) and reduce average claim rate α2(p1 , p2) , but the combined effect on the drift of the reserve difference in (10) is nevertheless of ambiguous sign, and further modelling indeed required. By (i) of Lemma 1, Vπ(δ) takes the form S(δ)/S(`u) , and combination of (ii) of the lemma and Remark 1allows characterizing a Stackelberg equilibrium with I2 as the leader and I1 the follower. It shows that the optimization problem is local: We can just consider maximization or minimization of κ(· , · ; δ) separately at each δ . This yields Proposition 1below, in which we find the explicit (local) conditions for a Stackelberg equilibrium in (11) in terms of the function κ(· , · ; δ) from (15) . For a solution to exist, the maximizing company should be facing a (locally, at least) concave problem structure, and the minimizing company a convex one. Existence cannot be guaranteed in general, but needs to be verified when considering a specific distribution of A , and hence a specific κ(· , · ; δ) . For the standard assumption of a gamma-distributed heterogeneity, we see in Section 5that an equilibrium does in fact exist and is unique. Although multiple solutions do not occur in this example, they cannot be excluded in general, so that the equilibrium may not be unique. The approach with backward induction should be the same, though giving a set of solutions. As multiple equilibria do not arise in the gamma case, we do not discuss them in more depth, except noting that uniqueness is guaranteed if the (local) concavity and convexity properties exploited in the following proposition extend globally. For a fixed δ , write d(p2) = b p1(δ|π2) for the optimal premium for I1 given I2 follows a strategy with premium p2at level δ. Proposition 1. In a Stackelberg equilibrium (π∗ 1 , π∗ 2) , the optimal set p∗ 1=p∗ 1(δ) , p∗ 2=p∗ 2(δ) of premiums at level δis a solution to p∗ 1=d(p∗ 2),p∗ 2=arg min p2 κ(d(p2),p2;δ),where d(p2) = arg sup p1 κ(p1,p2;δ). (16) The first order conditions for (p∗ 1,p∗ 2)are 0=κ0 1(p∗ 1,p∗ 2;δ) = κ0 2(p∗ 1,p∗ 2;δ), (17) Risks 2019,7, 49 15 of 23 Assuming for simplicity that the claim sizes are exponentially distributed with parameter θ , then we additionally have that zi=1 θe−θKi,z2 i=2 θ2e−θKi, for i=1, 2, together with excess risk ze 1,2 =E[min{Z,K1}−min{Z,K2}] = 1 θ(e−θK2−e−θK1). Aiming for an average claim size of 5000 e , we choose θ= 1 / 5000. We consider a deductible for I1 of 15% of the average claim size, that is K1= 750. Similarly for I2 with 10% of the claim size giving K2=500. Note in particularly that K1>K2. For these parameter values, we get z1=4303.54, z2 1=43 035 398.82, z2=4524.19, z2 2=45 241 870.90, ze 1,2 =220.65. (35) Assume further that there are N= 1, 000, 000 customers with identical personal safety loadings of ω(β) = 0.4 and that the risk-free interest rate is r= 3%. To get an indicator of the level of the reserves, we find a starting point, R , based on a 95% Value at Risk (VaR) principle. As N is rather large, the distribution of the sum ∑N/2 i=1(Zi− 5000 ) can be approximated by the normal distribution N(0, (N/2)/θ2). Solving for the Rthat satisfies PN/2 ∑ i=1 (Zi−5000)>R=0.05, using the inverse of the N ( 0, (N/ 2 )/θ2) cdf, yields R=5 815 435.77 . Next, I1 is assumed to have a reserve somewhat more than R, and I2somewhat less. More specifically, we let r0,1 = (1+γ)Rand r0,2 = (1−γ)R, (36) which leads to an initial reserve difference of δ= 2 γR . Choosing e.g., γ= 0.2 we get a difference of δ=2 326 174.31 . Since the analytic results do not depend on the bounds on the reserve, `u and `d , their particular values do not matter, and we just need that the interval [`d , `u] contains the chosen δ . Given this value, the graph of the criterion to be optimized, κ(p1 , p2 ; δ) , appears in Figure 2, and the corresponding contour diagram in Figure 3. Figure 2. Graph of κ(p1,p2;δ). Risks 2019,7, 49 16 of 23 250 300 350 400 450 p2 200 250 300 350 400 p1 Figure 3. Contour diagram of κ(p1,p2;δ). Recall that we here consider the case where I2 offers the better product ( K1>K2 ) and chooses its premium p2 first. Given this, I1 maximizes by seeking toward the ridge that appears diagonally when choosing p1 . The market leader, I2 , takes this response function of I1 into account, and minimizes κ(p1 , p2 ; δ) along the ridge, by choice of p2 . The optimum provides the Stackelberg equilibrium, at the saddle point. However, in this case the saddle is located diagonally, not parallel to the axes, and there is no Nash equilibrium. In particular, given p1 , I2 would benefit from increasing p2 , moving away from the ridge (toward cooler colors in the figures). While this precludes Nash equilibrium, the analysis demonstrates that it is possible to obtain an equilibrium in finite premiums by having I2 commit to some p2 at the given δ , then letting I1 respond, i.e., a Stackelberg equilibrium. This is also verified by the value D(a,b,K1,K2,r,δ,ω(β)) = −9603.91, which tells us that condition (32) is satisfied, whereas (34) is not, as − 4 ( 1 +ω(β)ze 1,2 =− 1235.62, i.e., greater than D(·)in this case. From Theorem 1, we compute the Stackelberg equilibrium premiums p∗ 1=305.5 and p∗ 2=326.0 (37) at the current reserve difference δ=2 326 174.31. These are to be compared with the net premiums α1(p∗ 1,p∗ 2)z1=0.0307 ·4303.54 =132.1, α2(p∗ 1,p∗ 2)z2=0.1693 ·4524.19 =766.0, so that pursuing solely the competition aspects would lead to a likely loss for I2 at the current reserve difference. This is not necessarily a paradox since the perspective of control and game theory is to focus solely on a one-eyed goal. Larger δ means I2 is lagging more behind the large firm I1 , and this gives I2 greater incentive to compete for customers by lowering its premium, with I1 responding by letting premiums move in lockstep. Thus, in equilibrium, I2 always receives a higher premium than I1 , reflecting the higher quality product (lower deductible). This type of product is attractive to “bad” customers, that is, customers with high claim frequency, as seen in Figure 4. These customers are expected to experience more losses than “good” customers, and are therefore willing to pay extra for better coverage, yielding a separating equilibrium, with customers’ choices revealing their type, as in Rothschild and Stiglitz (1976). Still, I2 may remain the smallest company, due to the higher risk of its customers. Risks 2019,7, 49 17 of 23 0mΓ Figure 4. Distribution of customers in equilibrium, where customers with claim frequencies in the green (blue) area insure at I1(I2). In Figures 5and 6, exhibiting aspects of D , we take a closer graphical look at the second order criteria. Starting with Figure 5, we plot D as a function of δ . All other parameters remain the same as above. Values for δ for which D<− 4 ( 1 +ω(β)ze 1,2 are plotted in green to indicate that the equilibrium is of Stackelberg type. Values that yield − 4 ( 1 +ω(β)ze 1,2 <D< 0, and hence equilibrium of Nash-type, are plotted in blue. Finally, the values plotted with red give D> 0, which tells us that there is no equilibrium. Here we see that D is indeed a linearly increasing function of δ , as it should be according to (30) and (33) . Hence, for small δ -values we get a Stackelberg equilibrium (green). For a small spectrum in the middle we get a Nash equilibrium (blue), and, finally, for large values of δ there is no equilibrium. The same color codes are used in Figure 6, which shows the color plateaus of D , and not the actual values, as depending on the deductibles, K1 and K2 . As we restrict the analysis to the case where K1>K2 , it is only the lower triangular part that is illustrated. For simultaneously large values (above 5 × 10 4 ) of K1 and K2 , there appears an area (red) where there is no equilibrium. However, 5 × 10 4 is ten times the average claim size of 5000 and obviously an unrealistically large value of the deductibles. For K1 and K2 being close, i.e., along the diagonal, there is then an equilibrium of Stackelberg type (green area) for smaller values. Moving away from the diagonal, the equilibrium type will change from Stackelberg to Nash (blue area). However, in the most realistic region of deductibles K1,K2being below the mean claim size 5000 =0.5 ×104it is always Stackelberg. Risks 2019,7, 49 18 of 23 0 5 10 15 ×1011 -1 -0.5 0 0.5 1 1.5 ×104 Figure 5. D as a function of δ . Green indicates Stackelberg equilibrium, blue indicates Nash equilibrium, and red indicates neither. Figure 6. D as a function of K1 and K2 for K1>K2 . Green indicates Stackelberg equilibrium, blue indicates Nash equilibrium, and red indicates neither. 6. Conclusions We have considered a non-life insurance market in which two insurance companies compete for customers by choice of premium strategies. Each company chooses its strategy to balance revenue against portfolio size, taking into account the strategy of the other company. We pay special attention to product differentiation and customer risk, while abstracting for simplicity from market frictions. For product differentiation, we focus on different deductibles, noting that alternatives would include bonus-malus systems, and proportional compensation in deductibles. The analysis is carried out in continuous time using stochastic differential game techniques. Adverse selection implies that a change in premium alters the risk composition of the portfolio. With claim arrival rates following a gamma distribution across customers, Stackelberg equilibrium premiums are derived. Conditions under which a Nash equilibrium exists are also established, but our numerical examples indicate that Stackelberg is the more typical case. Equilibrium premiums depend in an affine fashion on the running difference between the reserves of the companies, each modelled using the diffusion approximation to Risks 2019,7, 49 19 of 23 a standard Cramér-Lundberg risk process, extended to allow investment in a risk-free asset. Numerical illustrations of both types of equilibrium are provided. Overall, the managerial implications are that insurance companies should consider the premium as an active means to control portfolio size and revenue per customer in competition with other companies, as opposed to merely pooling individual risks and setting the premium based on conventional principles. Future research could consider three or more companies competing for market shares, to account explicitly for the risk of ruin or the possibility that some potential customers choose not to insure, or to pursue the more sophisticated ideas of Asmussen et al. (2019) on the customer’s problem. Author Contributions: Conceptualization, S.A., B.J.C., and J.T.; methodology, S.A. and J.T.; software, J.T.; validation, S.A., B.J.C., and J.T.; formal analysis, J.T.; writing–original draft preparation, S.A., B.J.C., and J.T.; writing–review and editing, B.J.C. and J.T. Funding: This research received no external funding Conflicts of Interest: The authors declare no conflict of interest. Appendix A. Second Order Derivative Tests for Theorem 1 Since we only need to consider a fixed δ , we write for notational convenience κ(p1 , p2) instead of κ(p1,p2;δ). Please note that the first order conditions, in the present case (17), can be written as κ0 i(p1,p2) = 1 κd(p1,p2)κ0 n,i(p1,p2)−κ(p1,p2;δ)κ0 d,i(p1,p2)=0 for i=1, 2, and consider the second order partial derivatives κ00 ii(p1,p2) = 1 κd(p1,p2)κ00 n,ii(p1,p2)−κ(p1,p2;δ)κ00 d,ii(p1,p2) −1 κd(p1,p2)κ0 d,i(p1,p2)κ0 n,i(p1,p2)−κ(p1,p2;δ)κ0 d,i(p1,p2) −1 κd(p1,p2)2κ0 d,i(p1,p2)κ0 n,i(p1,p2)−κ(p1,p2;δ)κ0 d,i(p1,p2), κ00 ij(p1,p2) = 1 κd(p1,p2) κ00 n,ij(p1,p2)−κ(p1,p2;δ)κ00 d,ij(p1,p2) −1 κd(p1,p2)κ0 d,i(p1,p2)κ0 n,j(p1,p2)−κ(p1,p2;δ)κ0 d,j(p1,p2)! −1 κd(p1,p2)2κ0 d,j(p1,p2)κ0 n,i(p1,p2)−κ(p1,p2;δ)κ0 d,i(p1,p2). In optimum the critical point (p∗ 1 , p∗ 2) must satisfy the first order condition (27) , which reduces the second order partial derivatives to κ00 ii(p∗ 1,p∗ 2) = 1 κd(p∗ 1,p∗ 2) κ00 n,ii(p∗ 1,p∗ 2)−˜ κκ00 d,ii(p∗ 1,p∗ 2)!, κ00 ij(p∗ 1,p∗ 2) = 1 κd(p∗ 1,p∗ 2) κ00 n,ij(p∗ 1,p∗ 2)−˜ κκ00 d,ij(p∗ 1,p∗ 2)!. Risks 2019,7, 49 20 of 23 From the links between the first order derivatives in the proof of Theorem 1, κ00 22(p∗ 1,p∗ 2) = 1 κd(p∗ 1,p∗ 2) κ00 n,22(p∗ 1,p∗ 2)−˜ κκ00 d,22(p∗ 1,p∗ 2)!(A1) =1 κd(p∗ 1,p∗ 2) κ00 n,11(p∗ 1,p∗ 2)−2f0 1(p∗ 1,p∗ 2)−˜ κκ00 d,11(p∗ 1,p∗ 2)!, (A2) κ00 12(p∗ 1,p∗ 2) = 1 κd(p∗ 1,p∗ 2) κ00 n,12(p∗ 1,p∗ 2)−˜ κκ00 d,12(p∗ 1,p∗ 2)!(A3) =1 κd(p∗ 1,p∗ 2) −κ00 n,11(p∗ 1,p∗ 2) + f0 1(p∗ 1,p∗ 2) + ˜ κκ00 d,11(p∗ 1,p∗ 2)!, (A4) where f(p1,p2) = γ(b,y/a)−Γ(b,y/a). The second order derivative test on the Hessian in (19), κ00 11(p∗ 1,p∗ 2)κ00 22(p∗ 1,p∗ 2)−κ00 12(p∗ 1,p∗ 2)2=−1 κd(p∗ 1,p∗ 2)2f0 1(p∗ 1,p∗ 2)2<0, then confirms a saddle point, provided we can show the condition (18) . For this, we need to be more specific and find the actual second order derivatives and evaluate them in equilibrium. Differentiating κ0 n,1(p1,p2)and κ0 d,1(p1,p2)with respect to p1yields κ00 n,11(p1,p2) = −2 a(1+ω(β))ze 1,2 e−y/a(y/a)b−1+z1+z2 a((1+ω(β))ze 1,2)2e−y/a(y/a)b−1(y/a−b) −p1+p2 (a(1+ω(β))ze 1,2)2e−y/a(y/a)b−2(y/a−b+1), κ00 d,11(p1,p2) = z2 2−z2 1 a((1+ω(β))ze 1,2)2e−y/a(y/a)b−1(y/a−b). Evaluating at the equilibrium premiums, κ00 n,11(p∗ 1,p∗ 2) = −2 a(1+ω(β))ze 1,2 e−mΓ/a(mΓ/a)b−1+z1+z2 a((1+ω(β))ze 1,2)2e−mΓ/a(mΓ/a)b−1(mΓ/a−b) −2p∗ 2−(1+ω(β))ze 1,2mΓ (a(1+ω(β))ze 1,2)2e−mΓ/a(mΓ/a)b−2(mΓ/a−b+1), κ00 d,11(p1,p2) = z2 2−z2 1 a((1+ω(β))ze 1,2)2e−mΓ/a(mΓ/a)b−1(mΓ/a−b). Multiplying by the positive constant κd(p∗ 1 , p∗ 2)a(( 1 +ω(β))ze 1,2)2/((mΓ/a)b−1exp(−mΓ/a)) , the criterion can be written in reduced form explicitly as −2(1+ω(β))ze 1,2 + (z1+z2)(mΓ/a−b)−(2p∗ 2/a−(1+ω(β))ze 1,2mΓ/a)(mΓ/a)−1(mΓ/a−b+1) −˜ κ(z2 2−z2 1)(mΓ/a−b)<0. Inserting the optimal premium, p∗ 2=a 21 2emΓ/a(mΓ/a)1−bΓ(b)(1+ω(β))ze 1,2 + (1+ω(β))ze 1,2(mΓ/a) + (mΓ/a)(z1+z2) −(mΓ/a)˜ κ(z2 2−z2 1), Risks 2019,7, 49 21 of 23 we can reduce the condition to ˜ κ(z2 2−z2 1)−2(1+ω(β))ze 1,2 −1 2emΓ/a(mΓ/a)−bΓ(b)(1+ω(β))ze 1,2(mΓ/a−b+1)−(z1+z2)<0, which is the same as (32). The condition (25) for a Nash equilibrium can also be found more explicitly by using the link in (A2) between the second order derivatives. The condition can be rewritten as 1 κd(p∗ 1,p∗ 2) κ00 n,11(p∗ 1,p∗ 2)−2f0 1(p∗ 1,p∗ 2)−˜ κκ00 d,11(p∗ 1,p∗ 2)!>0, which, using the same approach as above, can be written as 2a((1+ω(β))ze 1,2)2 (mΓ/a)b−1exp(−mΓ/a)f0 1(p∗ 1,p∗ 2)<˜ κ(z2 2−z2 1) −2(1+ω(β))ze 1,2 −1 2emΓ/a(mΓ/a)−bΓ(b)(1+ω(β))ze 1,2(mΓ/a−b+1)−(z1+z2), where a((1+ω(β))ze 1,2)2 (mΓ/a)b−1exp(−mΓ/a)f0 1(p∗ 1,p∗ 2) = −2a((1+ω(β))ze 1,2)2 (mΓ/a)b−1exp(−mΓ/a) (mΓ/a)b−1exp(−mΓ/a) a(1+ω(β)ze 1,2 =−2(1+ω(β))ze 1,2, which combined yields (34). References Albrecher, Hansjörg, Jozef L. 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