Game Theory Lecture 2

Game Theory Lecture 2 March 7, 2015 2 Cournot Competition Game and Transportation Game Nash equilibrium does not always occur in practice, due the imperfect information, bargaining, cooperation, sequential or dynamic decision, repeated decision, etc. However there are also many examples in which the behavior of players are indeed close to Nash Equilibrium. The Cournot oligopoly competition model and the traffic network model are two typical examples. 2.1 The KKT Optimality Condition We first introduce the so-called Karush Kuhn Tucker (KKT) optimality condition as technical preparation. Recall that if we want to minimize a function f(x 1,, x n ) then a necessary condition for its minimum point is: f(x 1,..., x n ) x i = 0, i = 1,..., n. The above result is essentially due to the 17th century French mathematician Pierre de Fermat. In the modern day notation, we write the condition as f(x ) = 0. (1) 1

Moreover, if function f( ) is convex, then condition (1) becomes sufficient. Basically the KKT condition deals with the following constrained problem: min f(x) s.t. h i (x) = 0, i = 1, 2,..., m. g j (x) 0, j = 1, 2,..., n. (2) Suppose the constraints above satisfies certain constraint qualification (for instance linear independent, slater condition, et. al.). Then if x is an optimal solution, there must exist λ i, i = 1,..., m, µ j 0, j = 1,..., r such that m r f(x ) + λ i h i (x ) + µ j g j (x ) = 0 and µ j h j (x ) = 0, j = 1,..., r. (3) i=1 j=1 Note that if problem (2) is convex, i.e. f(x), g j (x), j = 1,..., r are convex and h i (x), i = 1,..., m, are affine linear, then condition (3) is sufficient as well. 2.2 Cournot Oligopoly Competition Model This mode is first introduced by Antoine Augustin Cournot to describe the competition in the duopoly of spring water. The term Oligopoly represents a common market form in which only a few number of producers dominates the market. Duopoly is a market with two dominant players. This model assumes that there are n firms producing the same product. Each firm aims to maximize the total profit and decide the production level on its own after viewing the price of the product. Price of the product is assumed to be a decreasing function P (q) with respect to the total production level q = i q i. Each firm has its own production costc i (q i ) with respect to its own production level q i, which is assumed to be nonnegative and increasing. The profit of firm i is therefore a function of form: ( n ) u i (q i, q i ) = q i P q j C i (q i ). Let s start with the easy cases. If there are two firms with identical constant unit cost, i.e., C i (q i ) = cq i and linear price function P (q) = (a bq) +. Let s calculate the Nash equilibrium. 2 j=1

For firm 1, the profit is u 1 (q 1, q 2 ) = q 1 ((a bq 1 bq 2 ) + c) q 1 (a bq 1 bq 2 c), if q 1 a/b q 2 = cq 1, if q 1 a/b q 2. So the optimal response R 1 (q 2 ) = a c bq 2, if q 2b 2 a c b 0, if q 2 a c b. Similarly, for firm 2 the optimal response is R 2 (q 1 ) = a c bq 1, if q 2b 1 a c b 0, if q 1 a c b. Solving R 2 (R 1 (q 2 )) = q 2, we have q1 = q2 = a c, which is the only Nash equilibrium of this 3b game. Each firm now has profit of (a c)2, and the total profit is 2(a c)2. 9b 9b The above proof holds for any n firms with the same setting, in which the production level for each firm is q i = The price would be p = (a nbq i ) + = a+nc (n+1). a c (a c)2 n(a c)2, with profit of for each firm, and total profit of. (n+1)b (n+1) 2 b (n+1) 2 b We can see here that when the competition level increases, the price converges to the marginal cost, and the total profit level decreases with respect to the competition level. If there is an entry level of profit or fixed cost F for each firm to enter the market, the number n of firms should be approximately satisfy (a c) 2 (n + 1) 2 b = F, and consequently n = a c F F b 1, with production level q i at. b From Cournot model we can see that firms do have strong incentive to negotiate secrete contracts to reduce production level, and increase the overall profit. 3

2.3 Bertrand s Model of Oligopoly In this model, there are still n firms producing the same product, each with nonnegative and increasing production cost C i (q i ). The firms compete for customers, who buy from the firm which offers the lowest price, and if there are many firms who offer the same price they will choose one uniform randomly. The total demand with respect to the price level p is D(p), which is a nonnegative nonincreasing function. Firms now need to decide their price level and produce according to the realized demand that they receive. Take n = 2 case for example, and assume the firms have identical constant unit production cost, i.e., C i (q i ) = cq i, the demand is a linear function D(p) = a bp with b > 0. The utility function of firm i can be therefore represented as: p i D(p i ) C i (D(p i )), if p i < p i u i (p i, p i ) = 1 p 2 id(p i ) C i ( 1D(p 2 i)), if p i = p i 0 if p i > p i. In this game, each firm will try to marginally beat the other firm on the price, as long as the current price level is still strictly above c. So the only Nash equilibrium is (c, c), which means both firms have no profit at all. 2.4 Baress s Paradox A common intuition among most people is that if the more resources given, the better the outcome should be, however it is not the case in general. Sometimes more resources and options actually makes the system maybe even worse. Theoretically Baress s Paradox provides a simple example. Consider a traffic network from city A to city B, with middle point C and D. Originally, there are only two slow routes, ACB and ADB. Now because the traffic is heavy the government decides to build a one-way high way from point C to D. Is this decision wise to make or not? Although intuitively it looks like the high way would definitely improve the system overall, it is not always so. The following is directly quoted from Wikipedia: 4

In 1983 Steinberg and Zangwill provided, under reasnable assumptions, necessary and sufficient conditions for Braess s paradox to occur in a general transportation network when a new route is added. (Note that their result applies to the addition of any new route not just to the case of adding a single link.) As a corollary, they obtain that Braess s paradox is about as likely to occur as not occur; their result applies to random rather than planned networks and additions. In Seoul, South Korea, a speeding-up in traffic around the city was seen when a motorway was removed as part of the Cheonggyecheon restoration project. In Stuttgart, Germany after investments into the road network in 1969, the traffic situation did not improve until a section of newly-built road was closed for traffic again. In 1990 the closing of 42nd street in New York City reduced the amount of congestion in the area. In 2008 Youn, Gastner and Jeong demonstrated specific routes in Boston, New York City and London where this might actually occur and pointed out roads that could be closed to reduce predicted travel times. Let s consider the following example: Suppose the time of each driver spend on the lane l is a linear function c l (f l ) = a l f l + b l associated with the flow of drivers f l on the road. Each driver wants to minimize his/her own time spend on the road from A to B. Suppose there are n many identical drivers trying to travel from A to B. The unit cost (time) function of each edge is x, n, n, x and 0 respectively on AC, AD, CB, DB, and CD. Then if the high way route CD is not open, the obvious Nash Equilibrium (and unique) is half driver goes through C, and the other half goes through D, the total traveling time would be 1.5n 2. However if CD is opened, we can see that everyone will try to take the route ACDB, because that route takes less time than any other option. However, totally they spend 2n 2 time to travel. Therefore the system becomes worse as a whole, and it hurts driver individually too (since they are identical and have identical traveling time). It is also easy to notice that the Nash Equilibrium in the first case is actually optimal for the whole system, and that decision is also optimal for the social system even after route CD is opened. 5

2.5 Selfish Routing Model of Transportation Network We have seen in the above example that extra route may not lead to more efficient traffic system. Therefore it is important to identify when and how did it happen, and the ways to avoid it. Structures of the network have been therefore studied so as to avoid such pathological properties; see Korilis, Lazar and Orda [17]. Moreover, Azouzi, Altman and Pourtallier [8] gave specific guidelines to avoid the Braess paradox type of situations when upgrading the network. Besides the network topology design, Korilis, Lazar and Orda [16] also considered how to allocate link capacities such that the resulting Nash equilibria will be efficient, so as to avoid the paradoxical property. Akamatsu [1] focused on the dynamic framework using the notion of dynamic user equilibrium, where the Braess paradox may also arise. Later, Akamatsu and Heydecker [2] presented a generalization of the model. Recently, Lin and Lo [20] identified some good congestions in the dynamic setting, meaning that it can actually help eliminate the negative impact due to the Braess paradox at the equilibrium. Suppose that there are a set of resources E in a system. directed graph G = (V ; L) with the set of nodes V, and the set of arcs L (interchangeably we also use the terminology link as synonym of arc in this paper). Furthermore, we assume that V = n, L = m. Note that multiple parallel links are allowed but no self-loop exists. Let us denote A R n m to be the node-to-arc incidence matrix. 1 Suppose that there are K players in the game. Each player wishes to use a set of resources in the system. The demand can be splitting of the commodity is allowed). We denote the origin-destination (OD) pairs for all the players to be {s 1, d 1 }, {s 2, d 2 },..., {s K, d K }. Let r denote a vector in R K, where the component r k represents the amount of commodity that Player k needs to transport. The transportation plan of Player k will be given by a vector x k R m, which indicates the flow on each link. Clearly, a feasible flow is given by the constraints Ax k = r k δ s k r k δ d k, where the notation δ i signifies the unit vector in R n whose 1 Each row of A represents a node and each column of A represents an arc. For an arc connecting node i to node j, the corresponding column in A will have all 0 elements except for the i-th element, where it is +1, and the j-th element, where it is 1. 6

i-th component is one while all others are zero. For each link l, we denote the total flow on the link to be f l = K k=1 xk l. Moreover, let us denote the unit cost for the flow on the link l to be a function c l : f l c l (f l ). Therefore, the data (G, r, c) specifies an instance of the non-cooperative routing game that is of interest to us in this paper. Indeed, for Player k, given the decisions of other players (conventionally denoted as x k ) is to minimize his/her own transportation cost given as: C k (x k, x k ) = l L x k l c l (f l ). Naturally, given the decisions of all the players, the social cost is a simple summation: SC(x) = K k=1 Ck (x k, x k ). Let x Nash denote the flow when the game reaches a Nash equilibrium; i.e. a solution at which no player will be able to improve his/her situation unilaterally. At the same time, let us denote x to be the socially optimal solution the solution that minimizes the social cost function SC(x) over all feasible solutions. The socalled Price of Anarchy (PoA) is defined as: PoA = SC(xNash ). SC(x ) We shall first consider the case where the unit cost function is affine linear in the total flow; that is, the unit cost function on the link l is c l (f l ) = a l f l + b l, where a l, b l 0. Then, each player k will face the following optimization problem: (P k ) min l L (a lf l + b l )x k l s.t. Ax k = r k δ s k r k δ d k x k 0. Replacing f l with K k=1 xk l, the above problem for Player k is a convex quadratic program, in which the decision vector is x k : (P k ) min l L { [( ) ]} b l x k l + a l i k xi l x kl + (x kl )2 s.t. Ax k = r k δ s k r k δ d k x k 0. 7

Let y k R m be the Lagrangian multiplier associated with the equality constraint Ax k = r k δ s k r k δ d k. The Karush-Kuhn-Tucker optimality condition for (P k ) is: l L Axk = r k δ s k r k δ d k x k 0 b l + a l K i=1 xi l + a lx k l + (A T y k ) l z k l z k l 0, l = 1,..., m x k l zk l = 0, l = 1,..., m. = 0, l = 1,..., m Denote z k to be the vector whose l-th component is zl k, l = 1,..., m. A Nash equilibrium for the routing game is attained if and only if each player attains the optimum simultaneously; i.e., for all k = 1,..., K, we have l L Axk = r k δ s k r k δ d k x k 0 b + Diag(a) K i=1 xi + Diag(a)x k + A T y k z k = 0, z k 0, (x k ) T z k = 0, where Diag(a) is the diagonal matrix whose l-th diagonal is a l, l = 1,..., m. We can explicitly write the KKT optimality condition using the block matrix notation. Let x (respectively y, and z, and R) be the vector consisting of x 1,..., x K (respectively y 1,..., y K, and z 1,..., z K, and (r 1 δ s 1 r 1 δ d 1),..., (r K δ s K r K δ d K)) by stacking sequentially one on top of each other. The equations for the Nash equilibrium solutions are: (I k A)x = R e b + (E k Diag(a))x + (I k Diag(a))x + (I k A) T y z = 0, (NE) x 0, z 0, x T z = 0, where stands for the Kronecker product between two matrices, e is the (K by 1) all-one vector, E K is the (K by K) all-one matrix, and I K is the (K by K) identity matrix. 8

Again, by using KKT condition, we observe that the Nash equilibrium is the solution of the following optimization problem: (NEO) min (e b) T x + 1 2 xt ((E k + I k ) Diag(a)) x s.t. (I k A)x = R x 0. References [1] T. Akamatsu, A dynamic traffic equilibrium assignment paradox. Transportation Research Part B: Methodological, 34, pp. 515 531, 2000. [2] T. Akamatsu, B. Heydecker, Detecting Dynamic Traffic Assignment Capacity Paradoxes in Saturated Networks. Transportation Science, 37, No. 2, pp. 123 138, 2003. [3] S. Aland, D. Dumrauf, M. Gairing, B. Monien and F. Schoppmann, Exact price of anarchy for polynomial congestion games, In Proceedings of the 23rd Annual Symposium on Theoretical Aspects of Computer Science, 3884, pp. 218 229, 2006. [4] E. Altman and H. Kameda, Equilibria for multiclass routing in multi-agent networks. In Proceedings of the 40th Annual IEEE Conference on Decision and Control, 1, pp. 604 609, 2001. [5] E. Altman, T. Basar, T. Jimenez and N. Shimkin, Competitive routing in networks with polynomial cost, IEEE Trans. on Automatic Control, 47, pp. 92 96, 2002. [6] E. Altman, T. Boulogne, R. El Azouzi, T. Jimenez and L. Wynter, A survey on networking games in telecommunications, Computers & Operations Research, 33, pp. 286 311, 2006. [7] B. Awerbuch, Y. Azar, and L. Epstein, The price of routing unsplittable flow, In Proceedings of the thirty-seventh annual ACM Symposium on Theory of Computing, pp. 57 66, 2005. 9

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