Several strategic settings can be modeled as a tournament, whereby the probability of winning a certain prize not only depends on how much effort you exert, but also on how much effort other participants in the tournament exert. For instance, wars between countries, or R&D competitions between different firms in order to develop a new product, not only depend on a participant's own effort, but on the effort put by its competitors. Let's analyze equilibrium behavior in these settings. Consider that the benefit that firm 1 obtains from being the first company to launch a new drug is $36 million. However, the probability of winning this R&D competition against its rival (i.e., being the first to launch the drug) is which it increases with this firm's own expenditure on R&D, X1, relative to total expenditure, x1 + x2. Intuitively, this suggests that, while spending more than its rival, i.e., x > X2, increases firm l's chances of being the winner, the fact that x, > x, does not guarantee that firm 1 will be the winner. That is, there is still some randomness as to which firm will be the first to develop the new drug, e.g., a firm can spend more resources than its rival but be "unlucky" because its laboratory exploits a few weeks before being able to develop the drug. For simplicity, assume that firms' expenditure cannot exceed 25, i.e., x; [0,25). The cost is simply Xi, so firm l's profit function is 1, (x,x)=36 x x + x2 -X and there is an analogous profit function for country 2: 2 z(x,x)=30(w this 12 (x1,x2) = 36 x2 x + x x2 You can easily check that these profit functions are concave in a firm's own expenditure, i.e., Gx (x x). 50 for every firm i={1, 2} where jti. Intuitively, this indicates that, while profits increase Ox? in the firm's R&D, the first million dollar is more profitable than the 10th million dollar, e.g., the innovation process is more exhausted. a. Find each firm's best-response function. b. Find a symmetric Nash equilibrium, i.e., x;* = x; = x*. Several strategic settings can be modeled as a tournament, whereby the probability of winning a certain prize not only depends on how much effort you exert, but also on how much effort other participants in the tournament exert. For instance, wars between countries, or R&D competitions between different firms in order to develop a new product, not only depend on a participant's own effort, but on the effort put by its competitors. Let's analyze equilibrium behavior in these settings. Consider that the benefit that firm 1 obtains from being the first company to launch a new drug is $36 million. However, the probability of winning this R&D competition against its rival (i.e., being the first to launch the drug) is which it increases with this firm's own expenditure on R&D, X1, relative to total expenditure, x1 + x2. Intuitively, this suggests that, while spending more than its rival, i.e., x > X2, increases firm l's chances of being the winner, the fact that x, > x, does not guarantee that firm 1 will be the winner. That is, there is still some randomness as to which firm will be the first to develop the new drug, e.g., a firm can spend more resources than its rival but be "unlucky" because its laboratory exploits a few weeks before being able to develop the drug. For simplicity, assume that firms' expenditure cannot exceed 25, i.e., x; [0,25). The cost is simply Xi, so firm l's profit function is 1, (x,x)=36 x x + x2 -X and there is an analogous profit function for country 2: 2 z(x,x)=30(w this 12 (x1,x2) = 36 x2 x + x x2 You can easily check that these profit functions are concave in a firm's own expenditure, i.e., Gx (x x). 50 for every firm i={1, 2} where jti. Intuitively, this indicates that, while profits increase Ox? in the firm's R&D, the first million dollar is more profitable than the 10th million dollar, e.g., the innovation process is more exhausted. a. Find each firm's best-response function. b. Find a symmetric Nash equilibrium, i.e., x;* = x; = x*
