By Haurie A., Krawczyk J.
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Extra resources for An Introduction to Dynamic Games
E. 1) a value 0. 19). 19). We know that an equilibrium exists for a bimatrix game (Nash theorem). 19) with optimal value 0. Hence the optimal program (x∗ , y∗ , v∗1 , v∗2 ) must also give a value 0 to the objective function and thus be such that xT∗ Ay∗ + xT∗ By∗ = v∗1 + v∗2 . 18) xT Ay∗ ≤ v∗1 xT∗ By ≤ v∗2 . 42 CHAPTER 3. 20) imply xT∗ Ay∗ = v∗1 xT∗ By∗ = v∗2 . 18) xT Ay∗ ≤ xT∗ Ay∗ xT∗ By ≤ xT∗ By∗ and hence, (x∗ , y∗ ) is a Nash equilibrium for the bimatrix game. QED A Complementarity Problem Formulation We have seen that the search for equilibria could be done through solving a quadratic programming problem.
5. 5 51 Cournot equilibrium The model proposed in 1838 by [Cournot, 1838] is still one of the most frequently used game model in economic theory. It represents the competition between different firms supplying a market for a divisible good. 1 The static Cournot model Let us first of all recall the basic Cournot oligopoly model. We consider a single market on which m firms are competing. ,m qj is the total supply of goods on the market. The firm j faces a cost of production Cj (qj ), hence, letting q = (q1 , .
Qm ) represent the production decision vector of the m firms together, the profit of firm j is πj (q) = qj D(Q) − Cj (qj ). 1 The market demand and the firms cost functions satisfy the following (i) The inverse demand function is finite valued, nonnegative and defined for all Q ∈ [0, ∞). It is also twice differentiable, with D (Q) < 0 wherever D(Q) > 0. In addition D(0) > 0. (ii) Cj (qj ) is defined for all qj ∈ [0, ∞), nonnegative, convex, twice continuously differentiable, and C (qj ) > 0.
An Introduction to Dynamic Games by Haurie A., Krawczyk J.