Preliminaries

3.1.1 Terminology of game theory

Consider a game in which two players are participating. Each player’s preferences are captured by an objective function. However, the object function does not only depend on the decision of the player itself, but also on the strategy of the other.

If the players are able to enter a cooperative environment where binding decisions are done in conjunction, it is called as a cooperative game. However, we will exclusively consider non-cooperative games, where no such agreements are possible. A non- cooperative game is said to be zero-sum if one player’s gain is exactly balanced out with another player’s loss. Hence, available resources remain constant. Conversely, in a non-zero-sum game the aggregate gains and losses can be different from zero.

If the strategies of the players completely determine the outcome, the game is said to be deterministic. If, on the contrary, a random variable is influencing at least one of the object functions, the game is called stochastic. The game has complete information, if the players, their objective functions, and any underlying probability distribution are common information. If not so, the game is said to have incomplete information.

In a static game, the players only have access to these a priori information and move simultaneously. In a dynamic game, the players move sequentially or repeatedly, where some players are granted access to information about the previous decisions of others. A dynamic game is said to be a differential game if it plays out in continuous time, where a differential equation describing the state trajectory of the game is controlled over time by the decision processes of the players. This type of game is closely related to the theory of stochastic control.

3.1.2 Equilibrium types

Let x_i ∈ M_i be the decision variable of player i, where M_i is the set of possible actions. Let M=M₁ × M₂ ignoring that there might the coupled constraints on the decision variables.

Presume for the moment and without loss of generality that the objective functions of all players are value functions, which they seek to maximise. The value functions are denoted byV_i^(x1,x2) fori= 1,2. In case a player is a minimiser with a loss function L_i as the objective, it can transformed by V_i =−L_i into a maximisation problem. A Nash equilibrium is then defined as follows.

Definition 3.1 (Nash equilibrium). (x^∗₁, x^∗₂)∈ M is a Nash equilibrium if it satisfies V₁^{(x∗1,x∗2)} ≥V₁^(x1,x∗2) for all x₁ ∈ M₁

and

V^(x

∗ 1,x^∗₂)

2 ≥V^(x

∗ 1,x2)

2 for all x₂ ∈ M₂.

Hence, when at a Nash equilibrium neither of the players has the incentive to deviate.

If V = V₁ = −V₂ the game is of zero-sum type, where the same objective is maximised by Player 1 and minimised by Player 2. A Nash equilibrium (x^∗₁, x^∗₂) can then be defined equivalently as the saddle point

V^(x1,x∗2) ≤V^{(x∗1,x∗2)}≤V^(x∗1,x2) for all (x₁, x₂)∈ M. (3.1) This is the type of game and equilibrium considered in Paper C. Note that the sequence of which the players make their decisions is subordinate, since in this type of equilibrium we have

V^{(x∗1,x∗2)} = max

x1∈M₁ min

x2∈M₂V^(x1,x2) = min

x2∈M₂ max

x1∈M₁V^(x1,x2).

However, in a Stackelberg equilibrium the sequence of the game is indeed relevant.

Consider a sequential game where a leader makes the initial play. The follower observes the decision of the leader and then reacts. We define a Stackelberg-equilibrium in a zero-sum game where Player 2 is the (minimising) leader and Player 1 the (maximising) follower as:

Definition 3.2 (Stackelberg equilibrium). (x^∗₁, x^∗₂)∈ M is a Stackelberg equilibrium if

V^{(d(x∗2),x∗2)} ≤V^(d(x2),x2) for all x₂ ∈ M₂ where

d(x₂) = {ς ∈ M₁ :V^(ς,x2)≥V^(x1,x2) for all x₁ ∈ M₁}.

It is solved by backward induction, where the follower finds the optimal response as a function of leader’s decision. The leader insert this information about the response function into the value function and solves for the optimal opening play. In Stackelberg equilibrium, the value function is

V^{(x∗1,x∗2)} = max

x1∈M₁ min

x2∈M₂V^(x1,x2)

However, unlike the Nash equilibrium, the maximum and minimum in a Stackel- berg equilibrium can not be interchanged. In Paper Da sequential zero-sum game is considered in which we examine the existence of a Stackelberg (and a Nash) equilibrium.

Chapter 3. Equilibrium premium strategies for push-pull competition

3.1.3 A stochastic differential game

We now intend to extend the formulation of Section 2.1.5 to involve two insurance companies. Insurance Company i has to decide upon a strategy u_i = (u_i,t)t≥0 taking values in some admissible set Ui ⊆R. The reserve is then assumed to be governed by the strategies (u₁, u₂) as follows

dR^(u_i,t^1,u2) =µ_i(R^(u_1,t^1,u2), R^(u_2,t^1,u2), u_1,t, u_2,t)dt +σ_i(R^(u_1,t^1,u2), R^(u_2,t^1,u2), u_1,t, u_2,t)dW_i,t

withR^(u_i,0^1,u2)= ri,0fori, and(W1,t)t≥0 and(W2,t)t≥0 are independent Wiener processes.

We here generalise the objective function of Section2.1.5 by V^(u1,u2)(t, r₁, r₂) =E

hZ τ

t

e^−ζ(s−t)v(R^(u_1,s^1,u2), R^(u_2,s^1,u2), u_1,s, u_2,s)ds + e^{−ζ(τ−t)}K(τ, R^(u_1,τ^1,u2), R^(u_2,τ^1,u2))

R^(u_1,t^1,u2)=r₁, R^(u_2,t^1,u2) =r₂, t < τi ,

(3.2)

where v : R×R×U₁×U₂ → R, K : R×R → R, τ is an exit time, and ζ > 0 a discounting parameter. In a zero-sum setting as considered in the previous section, Player 1 chooses its strategy u1 in order to maximise V^(u1,u2) and Player 2 chooses its strategy u₂ to minimise it.

This is the type of competition studied in the papers of the present chapter, where the control at an insurance company’s disposal is the single customer premium.

The exit time is defined as τ = τ(δ) = inf{t >0 : R_1,t^(u1,u2)−R^(u_2,t^1,u2) ∈/ [`<sub>d</sub>, `_u]|δ = R_1,0−R_2,0 > 0}, where the upper and lower value of the interval involved must satisfy that δ ∈ [`d, `u]. Insurance Company 1 (the larger one based on initial capital) then chooses it’s policy premium to maximise the probability that the reserve difference hits the upper barrier before it hits the lower. Insurance Company 1 wants to push its competitor even further away. Conversely, Insurance Company 2 (the smaller one) chooses it’s policy premium to minimise the same probability, i.e. it wants pull closer to Insurance Company 1. The value function is then (3.2) with v(R^(u_1,s^1,u2), R^(u_2,s^1,u2), u_1,s, u_2,s) = 0 for all inputs, ζ = 0, and K(τ, R_1,τ, R_2,τ) = 1{R1,τ−R2,τ=`u}.

References

Bensoussan, A. and Friedman, A. (1977). “Nonzero-sum stochastic differential games with stopping times and free boundary problems”. Transactions of the American Mathematical Society 231.2, pp. 275–327.

Bensoussan, A., Siu, C. C., Yam, S. C. P., and Yang, H. (2014). “A class of non-zero- sum stochastic differential investment and reinsurance games”. Automatica 50.8, pp. 2025–2037.

Bertrand, J. (1883). “Théorie mathématique de la richesse sociale”. Journal des savants 67.1883, pp. 499–508.

Chen, S., Yang, H., and Zeng, Y. (2018). “Stochastic differential games between two insurers with generalized mean-variance premium principle”.ASTIN Bulletin 48.1, pp. 413–434.

Cournot, A. A. (1838). Recherches sur les principes mathématiques de la théorie des richesses. L. Hachette.

Elliott, R. J. and Davis, M. (1981). “Optimal play in a stochastic differential game”.

SIAM Journal on Control and Optimization 19.4, pp. 543–554.

Elliott, R. J. (1977). “The existence of optimal strategies and saddle points in stochastic differential games”. Differential Games and Applications (Lecture Notes in Control and Information Sciences). Vol. 3. Springer, Berlin, Heidelberg, pp. 123–135.

Elliott, R. J. and Siu, T. K. (2011a). “A BSDE approach to a risk-based optimal investment of an insurer”.Automatica 47.2, pp. 253–261.

— (2011b). “A stochastic differential game for optimal investment of an insurer with regime switching”. Quantitative Finance 11.3, pp. 365–380.

Jin, Z., Yin, G., and Wu, F. (2013). “Optimal reinsurance strategies in regime- switching jump diffusion models: Stochastic differential game formulation and numerical methods”. Insurance: Mathematics and Economics 53.3, pp. 733–746.

Nash, J. (1951). “Non-cooperative games”. Annals of mathematics 54.2, pp. 286–295.

Ramachandran, K. M. and Tsokos, C. P. (2012). Stochastic differential games. Theory and applications. Vol. 2. Springer Science & Business Media.

Taksar, M. and Zeng, X. (2011). “Optimal non-proportional reinsurance control and stochastic differential games”.Insurance: Mathematics and Economics 48.1, pp. 64–71.

Taylor, G. C. (1986). “Underwriting strategy in a competitive insurance environment”.

Insurance: Mathematics and Economics 5.1, pp. 59–77.

von Neumann, J. and Morgenstern, O. (1947). Theory of games and economic behavior. 2nd. Princeton University Press.

von Stackelberg, H. (1934). Marktform und Gleichgewicht. J. Springer.

Zeng, X. (2010). “A stochastic differential reinsurance game”. Journal of Applied Probability 47.2, pp. 335–349.

Paper C

Nash equilibrium premium strategies for push-pull competition in a

frictional non-life insurance market

søren asmussen, bent jesper christensen and julie thøgersen

abstract. Two insurance companiesI₁, I₂ with reserves R_1,t, R₂1, t compete for customers, such that in a suitable stochastic differential game the smaller company I2 with R2,0 < R1,0 aims at minimising R1,t − R2,t by using the premium p2 as control and the larger I₁ at maximising by using p₁. The dependence of reserves on premia is derived by modelling the customer’s problem explicitly, accounting for market frictions V, reflecting differences in cost of search and switching, information acquisition and processing, or preferences. AssumingV to be random across customers, the optimal simultaneous choice p^∗₁, p^∗₂ of premiums is derived and shown to provide a Nash equilibrium for beta distributed V. 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; diffusion approximation; exit problem;

market friction; Nash equilibrium; saddle point; beta distribution

Publication details: Insurance: Mathematics and Economics, 2019, vol. 87, pp. 92–100.

DOI: 10.1016/j.insmatheco.2019.02.002