EXAMPLE 176.2 (A synergistic relationship) Consider a variant of the situation in Example 39.1, in which two individuals are involved in a synergistic relationship. Suppose that the players choose their effort levels sequentially, rather than simul- taneously. First individual 1 chooses her effort level a₁, then individual 2 chooses her effort level a2. An effort level is a nonnegative number, and individual i's pref- erences (for i= 1, 2) are represented by the payoff function a; (c +a; - a;), where j is the other individual and c> 0 is a constant. To find the subgame perfect equilibria, we first consider the subgames of length
EXAMPLE 176.2 (A synergistic relationship) Consider a variant of the situation in Example 39.1, in which two individuals are involved in a synergistic relationship. Suppose that the players choose their effort levels sequentially, rather than simul- taneously. First individual 1 chooses her effort level a₁, then individual 2 chooses her effort level a2. An effort level is a nonnegative number, and individual i's pref- erences (for i= 1, 2) are represented by the payoff function a; (c +a; - a;), where j is the other individual and c> 0 is a constant. To find the subgame perfect equilibria, we first consider the subgames of length
Chapter1: Making Economics Decisions
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Transcribed Image Text:EXAMPLE 176.2 (A synergistic relationship) Consider a variant of the situation in
Example 39.1, in which two individuals are involved in a synergistic relationship.
Suppose that the players choose their effort levels sequentially, rather than simul-
taneously. First individual 1 chooses her effort level a₁, then individual 2 chooses
her effort level a2. An effort level is a nonnegative number, and individual i's pref-
erences (for i = 1, 2) are represented by the payoff function a¡(c+a; − a;), where j
is the other individual and c> 0 is a constant.
To find the subgame perfect equilibria, we first consider the subgames of length
1, in which individual 2 chooses a value of a2. Individual 2's optimal action after
the history ay is her best response to a₁, which we found to be. (c + a₁) in Exam-
ple 39.1. Thus individual 2's strategy in any subgame perfect equilibrium is the
function that associates with each history a₁ the action (c+a₁).
Now consider individual 1's action at the start of the game. Given individ-
ual 2's strategy, individual 1's payoff if she chooses a₁ is a₁(c +(c + a₁) -a₁),
or a₁ (3c-a₁). This function is a quadratic that is zero when a₁ = 0 and when
a₁ = 3c, and reaches a maximum in between. Thus individual 1's optimal action
at the start of the game is a₁ = c.
. We conclude that the game has a unique subgame perfect equilibrium, in which
individual 1's strategy is a₁ = 3c and individual 2's strategy is the function that
associates with each history at the action (c+a₁). The outcome of the equilibrium
is thåt individual 1 chooses a₁ = 3c and individual 2 chooses a2 = c.
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