# Competition model under mean-variance risk aversion

Consider the risk-averse behaviour of carrier 2 is measured by the mean- variance criterion. The utility functions of the two carriers can be defined as *U = E* (я^{-}]), and *U _{2} = E (n_{2})* — k-Std (ж

_{2}), where /•(.) is the mathematical expectation, and Std(.) takes the standard deviation. From (5.1)—(5.3), we have:

In this scenario, carrier 1 aims to maximise the expected profit. Carrier 2 is interested in maximising its mean profit minus the variance multiplied by a constant *X.* The parameter *X* represents the degree of risk aversion. Namely, the larger *X* is, the more risk-averse the carrier is. Hence, the utility of the agent increases with the mean profit and decreases with the variance. When rate of decrease with the variance is larger, the more risk-averse the carrier is.

Since the utility functions L/j(.) and *U*_{2}(.) are quadratic functions of *p]* and *p _{2},* respectively, they are concave in

*p*and

*p*The duopoly game on pricing decisions has a unique Nash equilibrium. By Finding the best response functions using the first-order conditions, the unique Nash equilibrium for the pricing game can be obtained.

_{2}.**Proposition 5.1**

Under the mean-variance risk-aversion criterion, the Nash equilibrium solution to the pricing game is given by:

where B| = (1 — *в)Е(%) + <Хро* + /Vi> and ^2 ^{=} *@E(4) ^{+ a}iPo* +

*P*

*2*

^{C}*2*

*-*Proof: the first-order conditions of (5.4) and (5.5) yield:

Solve the above two equations with respect to *p* and and ^{we can} obtain the results. This completes the proof

Proposition 5.1 provides the closed-form of the equilibrium prices of two carriers. Plug them into (5.4) and (5.5), we can obtain the closed-form of the optimal utility functions for two carriers. From Proposition 5.1, it is easy to derive how the equilibrium prices are responding to the benchmark shipping price *po,* the unit cost of each carrier *q,* and the risk-averse indicator Я as follows.

**Proposition 5.2**

Under the mean-variance risk-aversion criterion, (i) the equilibrium prices for two carriers are increasing in the benchmark shipping price *p*о and the unit cost of each carrier *q;* (ii) the equilibrium prices are decreasing in the risk-averse indicator Я.

The results in Proposition 5.2(i) reflect the intuition that higher benchmark shipping prices or higher unit costs of the shipping service would enable carriers to set up higher spot market prices. The results in Proposition 5.3(ii) can be interpreted as follows. When carrier 2 is more risk-averse, it is more conservative about the demand uncertainty. As a result, it tends to believe that the demand may be smaller and then sets the price lower to mitigate the risk. In response to carrier 2’s lower price, carrier 1 has to decrease its price to keep it competitive.

**Proposition 5.3**

Assuming that two carriers are symmetric in terms of oq = *Ob.* = *CC, Y* = *Yi ^{=}* 7> and Cj

*=C*

*2*

*=c.*Let

*po =*0. Then, the equilibrium prices can be simplified as (where /3 =

*a + Y),*

170 *Vessel logistics and shipping operations *
Proposition 5.4

Assuming that two carriers are symmetric with *a=Ot _{2}= Cl, Y=Y*

*2*

^{=}Y>^{anc}^ Cl

*=c*Let

_{2}=c*p*0. Then,

_{()}=- (i) The carrier l’s equilibrium price is decreasing in
*9.* - (ii) The carrier 2’s equilibrium price is increasing in
*9*if (2/3 — /)£(£)> 2/ЗА*Std*(9.

Proposition 5.4(i) indicates that carrier 1 will decrease its price to attract more demand as its market share decreases, which is in agreement with intuition. Proposition 5.4(h) reveals if carrier 2 is highly risk-averse such that (2/? — *y)-E *(<*) < 2/tt-Std (£), then it will decrease its price as its market share increases. This can be explained by the fact that when carrier 2’s market share *в* increases, the variance of its demand is also increasing. As a highly risk-averse player, carrier 2 tends to reduce prices to avoid a big variation of the profit.