In many situations, we are confronted with the decision of whether to challenge someone who is currently engaged in a particular activity. In personal relationships, for instance, we decide whether it is worthwhile to push our own agenda over that of a partner; in business, potential new firms have to decide whether to challenge an incumbent firm (as discussed in one of the examples in the text); and in elections, politicians have to decide whether to challenge incumbents in higher level electoral competitions.
A: Consider the following game that tries to model the decisions confronting both challenger and incumbent: The potential challenger moves first — choosing between staying out of the challenge, preparing for the challenge and engaging in it, or entering the challenge without much preparation. We will call these three actions O (for “out”), P (for “prepared entry”) and U (for “unprepared entry”). The incumbent then has to decide whether to fight the challenge (F) or give into the challenge (G) if the challenge takes place; otherwise the game simply ends with the decision of the challenger to play O.
(a) Suppose that the payoffs are as follows for the five potential combinations of actions, with the first payoff indicating the payoff to the challenger and the second payoff indicating the payoff to the incumbent: (P,G) leads to (3,3); (P,F) leads to (1,1); (U,G) leads to (4,3); (U,F) leads to (0,2); and O leads to (2,4). Graph the full sequential game tree with actions and payoffs.
(b) Illustrate the game using a pay off matrix (and be careful to account for all strategies).
(c) Identify the pure strategy Nash equilibria of the game and indicate which of these is sub game perfect.
(d) Next, suppose that the incumbent only observes whether or not the challenger is engaging in the challenge (or staying out) but does not observe whether the challenger is prepared or not. Can you use the logic of sub game perfection to predict what the equilibrium will be?
(e) Next, suppose that the payoffs for (P,G) changed to (3,2), the payoffs for (U,G) changed to (4,2) and the payoffs for (U,F) changed to (0,3) (with the other two payoff pairs remaining the same). Assuming again that the incumbent fully observes both whether he is being challenged and whether the challenger is prepared, what is the sub game perfect equilibrium?
(f) Can you still use the logic of sub game perfection to arrive at a prediction of what the equilibrium will be if the incumbent cannot tell whether the challenger is prepared or not as you did in part (d)?
B: Consider the game you ended with in part A(f).
(a) Suppose that the incumbent believes that a challenger who issues a challenge is prepared with probability δ and not prepared with probability (1−δ). What is the incumbent’s expected payoff from playing G? What is his expected payoff from playing F?
(b) For what range of δ is it a best response for the incumbent to play G? For what range is it a best response to play F?
(c) What combinations of strategies and (incumbent) beliefs constitute a pure strategy sub game perfect equilibrium? (Be careful: In equilibrium, it should not be the case that the incumbent’s beliefs are inconsistent with the strategy played by the challenger!)
(d) Next, suppose that the payoffs for (P, G) changed to (4, 2) and the payoffs for (U, G) changed to (3, 2) (with the remaining payoff pairs remaining as they were in A (f)). Do you get the same pure strategy sub game perfect equilibria?
(e) In which equilibrium—the one in part (c) or the one in part (d)—do the equilibrium beliefs of the incumbent seem more plausible?