Game Theory in a Duooly The Cournot Model How does the outcome of cometition among two Internet Service Providers (ISPs) in a cometitive market deend on the characteristics of the demand for the ISPs' outut, the nature of the ISPs' cost functions, and the number of ISPs? Will the benefits of technological imrovements be assed on to consumers? Will a reduction in the number of ISPs generate a less desirable outcome? To answer these questions we need a model of the interaction between ISPs cometing for the business of consumers. Now such a situation based on the Cournot model is analyzed Key assumtion ios the oligololy situation which means that a cometition between a small number of ISPs will comete about a relatively big ammount of customers. Basic assumtion is the Cournot Model. More detailed information can be found on htt://mba.vanderbilt.edu/mike.shor/courses/game-theory/docs/lectures0456/ oligooly.html N ISPs comete on the market and offer their service to the customers. The cost to ISP i of roducing qi units of the good is Ci(qi), where Ci is an increasing function (more outut is more costly to roduce). In this examle the good is bandwidth. All the outut is sold at a single rice, determined by the demand for the good and the ISPs' total outut. Secifically, if the ISPs' total outut is Q then the market rice is P(Q); P is called the inverse demand function. Assume that P is a decreasing function when it is ositive: if the ISPs' total outut increases, then the rice decreases (unless it is already zero). If the outut of each ISP i is qi, then the rice is P(q1 + + qn), so thatisp i's revenue is qip(q1 + + qn). Thus ISP i's rofit, equal to its revenue minus its cost, is Fi(q1,..., qn) = qip(q1 + + qn) - Ci(qi). Cournot suggested that the industry be modeled as the following strategic game: Players: The ISPs. Actions: Each ISP's set of actions is the set of its ossible oututs (nonnegative numbers). Preferences: Each ISPs references are reresented by its rofit. For secific forms of the functions Ci and P a Nash equilibrium of Cournot's game can be comuted. Suose there are two ISPs (the industry is a duooly ), each ISP's cost function is the same, given by Ci(qi) = cqi for all qi ( unit cost is constant, equal to c), and the inverse demand function is linear where it is ositive, given by P(Q) = a - Q if Q <= a 0 if Q > a, where a > 0 and c >= 0 are constants. This inverse demand function is shown in the above figure. (Note that the rice P(Q) cannot be equal to a - Q for all values of Q, for then it would be negative for Q > a.)
Assume that c < a, so that there is some value of total outut Q for which the market rice P(Q) is greater than the ISPs' common unit cost c. (If c were to exceed a, there would be no outut for the ISPs at which they could make any rofit, because the market rice never exceeds a.) To find the Nash equilibria in this examle, we can use the rocedure based on the ISPs' best resonse functions. First we need to find the ISPs' ayoffs (rofits). If the ISPs' oututs are q1 and q2 then the market rice P(q1 +q2) is a - q1 - q2 if q1 + q2 >= a and zero if q1 + q2 > a. Thus ISP 1's rofit is F1(q1, q2) = q1(p(q1 + q2) - c) = q1(a - c - q1 - q2) if q1 + q2 <= a = -cq1 if q1 + q2 > a. In the following, one traditional demand function as shown in the following figure is assumed. Price (Q) a Demand Offer 1 o 2 q q q q q 3 1 o 2 4 a Quantity q To find ISP1's best resonse to any given outut q2 of ISP2, we need to study ISP1's rofit as a function of its outut q1 for given values of q2. If q2 = 0 then ISP1's rofit is F1(q1, 0) = q1(a - c - q1) for q1 <= a, a quadratic function that is zero when q1 = 0 and when q1 = a - c. This function is the right curve in the next figure.
F(q1,q2) (a-c-q2)/2 (a-c-q2) a Given the symmetry of quadratic functions, the outut q1 of ISP1 that maximizes its rofit is q1 = 1/2 (a - c). (If you know calculus, you can reach the same conclusion by setting the derivative of ISP1's rofit with resect to q1 equal to zero and solving for q1.) Thus ISP1's best resonse to an outut of zero for ISP 2 is b1(0) = 1/2 (a - c). As the outut q2 of ISP 2 increases, the rofit ISP1 can obtain at any given outut decreases, because more outut of ISP 2 means a lower rice. The left curve in the above figure is an examle of F1(q1, q2) for q2 > 0 and q2 < a - c. Again this function is a quadratic u to the outut q1 = a - q2 that leads to a rice of zero. Secifically, the quadratic is F1(q1, q2) = q1(a - c - q2 - q1), which is zero when q1 = 0 and when q1 = a c - q2. From the symmetry of quadratic functions (or some calculus) we conclude that the outut that maximizes F1(q1, q2) is q1 = 1/2 (a-c-q2). (When q2 = 0, this is equal to 1/2, the best resonse to an outut of zero that we found in the revious aragrah.) When q2 > a - c, the value of a - c - q2 is negative. Thus for such a value of q2, we have q1(a - c - q2 - q1) < 0 for all ositive values of q1: ISP1's rofit is negative for any ositive outut, so that its best resonse is to roduce the outut of zero. We conclude that the best resonse of ISP 1 to the outut q2 of ISP 2 deends on the value of q2: if q2 <= a-c then ISP1's best resonse is 1/2 (a-c-q2), whereas if q2 > a - c then ISP1's best resonse is 0. Or, more comactly, b1(q2) = 1/2 (a - c - q2) if q2 <= a c 0 if q2 > a - c.
Because ISP 2's cost function is the same as ISP 1's, its best resonse function b2 is also the same: for any number q, we have b2(q) = b1(q). Of course, ISP2's best resonse function associates a value of ISP2's outut with every outut of ISP1, whereas ISP1's best resonse function associates a value of ISP1's outut with every outut of ISP2, so we lot them relative to different axes. They are shown in the following figure (b1 is the steeer one, b2 is the less steeer one). /3 /3 As for a general game, b1 associates each oint on the vertical axis with a oint on the horizontal axis, and b2 associates each oint on the horizontal axis with a oint on the vertical axis. A Nash equilibrium is a air (q1*, q2*) of oututs for which q1* is a best resonse to q2*, and q2* is a best resonse to q1*: q1* = b1(q2*) and q2* = b2(q1*) The set of such airs is the set of oints at which the best resonse functions in the figure above intersect. From the figure we see that there is exactly one such oint, which is given by the solution of the two equations q1* = 1/2 (a - c - q2) q2* = 1/2 (a - c - q1).
Solving these two equations (by substituting the second into the first and then isolating q1, for examle) we find that q1* = q2* = 1/3 (a - c). In summary, when there are two ISPs, the inverse demand function is given by P(Q) = a - Q for Q <= a, and the cost function of each ISP is Ci(qi) = cqi, Cournot's oligooly game has a unique Nash equilibrium (q1*, q2*) = (1/3 (a - c), 1/3 (a - c)). The total outut in this equilibrium is 2/3, so that the rice at which outut is sold is P( 2/3 (a - c)) = 1/3 (a + 2c). As a increases (meaning that consumers are willing to ay more for the good), the equilibrium rice and the outut of each ISP increases. As c (the unit cost of roduction) increases, the outut of each ISP falls and the rice rises; each unit increase in c leads to a two-thirds of a unit increase in the rice.