10. Your little twin sisters (whom you lovingly refer to as Thing 1 and Thing 2) are driving you crazy! You’ve baked them a lovely birthday cake, but they won’t stop fighting over who gets the biggest slice. To settle the dispute, you draw on a time-honored ritual: You ask Thing 1 to cut the cake, and Thing 2 to choose which piece she wants.

a. Draw the extensive form of this game. Let Thing 1’s strategies be “Cut Evenly” or “Cut Unevenly”; depending on what’s on the platter, Thing 2’s strategies might include “Take Big Slice,” “Take Small Slice,” or “Take Equal Slice.” Assign payoffs to Thing 1 and Thing 2 that grow with the size of the slice they receive.

b. Use backward induction to find the equilibrium outcome for this game. Is the equilibrium consistent with your experience?

c. After the rules are announced, Thing 2 says,

“It’s not fair! I want to be the one who gets to cut the cake, not the one that chooses the slice!” Is Thing 2’s complaint valid — in other words, is there a first-mover advantage in this game?