Question

In many countries, armed forces rely both on volunteers and conscription for military service. For example, in Denmark, eligible men for military service are required to draw a number. Depending on the number they draw, they may get drafted if there aren't enough volunteers.

Consider a very simple game with two risk-neutral players, who are eligible for military service.[1]The Army needs only one of them.

Player 1 moves first, and decides whether or not to volunteer.

If Player 1 volunteers, then the game ends. Player 1 receives a payoff of (B -C), where B represents the benefit of volunteering and C represent the cost of joining the army, respectively. (You can think of both monetary and non-monetary benefits and costs). In this case, Player 2 gets a payoff of zero.

If Player 1 does not volunteer, then Player 2, who observes this decision, decides whether or not to volunteer.

Similarly, if Player 2 volunteers, the game ends with leaving Player 2 a payoff of (B -C), whereas Player 1 obtains a payoff of 0.  If Player 2 does not volunteer either, then the army drafts one of the players with the luck of the draw. Therefore, Player 1 and Player 2 gets drafted with an equal probability. 0.5. As the result of the draw, the player who ends up being drafted receives a payoff of –C (since the benefits of volunteering does not accrue in this case). The player who ends up not being drafted receives a payoff of 0.

a) I suggest for you to start by drawing the game tree, using Nature as a non-strategic player representing the (possible) uncertainty in the game. You do NOT need to turn this tree in, but it will be helpful to sketch it out.

b)    Let B=400 and C=600. What is the rollback equilibrium of this game? [10 points]

c)     Does this game with the payoff structure described in (b) exhibit first-mover advantage, second-mover advantage, or neither? Explain. [10 points]

d)    Now consider that Player 2 has a different value of C than Player 1.  Determine the minimum value of C, which makes the following commitment credible: “I will not volunteer regardless of what you do.” [10 points]

e)     Choose a larger value of C than you found in part (d), and solve the game for rollback equilibrium. Is the equilibrium outcome different than the one you have described in part (b)? [10 points]

Answer 1

(b)   Let equilibrium pay-off of player 1 & player 2 be (p1, p2) (see tree graph below)

If B = 400 and C = 600,

and player 1 has the first option, so he will consider his best response after considering the best response of player 2

For Player 2, if he decides Yes, his net payoff is -200

But if he decides no, his expected net pay-off is -300 (50% chance of -600 if he is selected and 50% chance of 0 if he is not selected)

Since -200 > -300, so, player 2 is better off by saying yes

Now, for player 1, if he says yes, his net pay-off is -200

And if he says no and player 2 says yes, then his net pay-off is 0

Hence, if player 1 knows that player 2’s best response is to say yes, then player 1’s best response is to say no. Thus, the rollback equilibrium of this game is Player 1 saying No and Player 2 saying Yes with their net payoffs being (0, -200) or, (p1, p2) = (No, Yes) = (0, -200)

(c) Yes, this game with the given payoff structure exhibits first-mover advantage as the payoff of the player 1 is better than that of player 2 in the rollback equilibrium.

If the player 2 had the chance to decide first than he would have the similar opportunity to consider the best response of player 1 before deciding his best response.

The pay-offs of player 1 and player 2 would subsequently reverse as the best response of player 1 would be to say yes and the best response of player 2 would be to say no with the net payoffs of player 1 being -200 and player 2 being 0.

or, (p1, p2) = (Yes, No) = (-200,0)

(d) If C of player 1 is C1 and C of player 2 is C2 where C1 ≠ C2 and If player 2 commits by saying that he will not say yes, then his commitment will be considered credible by player 1 only if C2 has the following value:

Net payoff of player 2 in saying Yes is less than net expected payoff by saying no

Or, 400-C2 <-300

Or, C2 > 700

Thus, the minimum value of C2 should be more than 700 for his commitment to be considered credible

e) suppose C2 > 700, let C2 be 800

then, player 2 is better off by saying no as his net pay-off is (400-800) = -400 if he said yes but his expected net pay-off would have been -300 if he said no

now, if player 1 knows that best response of player 2 is to say no, then in that case the expected net payoff of player 1 would be 0.5*(-600) + 0.5*(0) = -300

But if player 1 said yes in his first chance of choice, then his net payoff would be -200

Since – 200 > -300, so player 1 is better off by saying yes

Now, if the best response of player 1 is to say yes, then net payoff of player 2 is 0

Hence, the rollback equilibrium in this case is player 1 saying yes, player 2 no (by default) and the net payoff of player 1 is -200 and player 2 is 0

Yes the rollback equilibrium found here: (p1, p2) = (Yes, No) = (-200, 0) is different and opposite to rollback equilibrium found in case (b) (p1, p2) = (No, Yes) = (0, -200)