$3.$ Consider the following "constant sum" centipede game:

$-$ Player one moves first, and can choose stop or continue. If he plays stop, player one gets a payoff of $6$ and player two gets a payoff of $6.$ If he plays continue, player two faces the following choice:

$-$ Player two can choose stop or continue. If she chooses stop, player one gets a payoff of $5$ and player two gets a payoff of $7.$ If she chooses continue, the game ends and player one gets a payoff of $8$ and player two gets a payoff of $4.$

a$)$ Illustrate this game in extensive form $($i$.$e$.$ as a game tree$),$ and find the Subgame Perfect Equilibrium.

b$)$ Levitt, List, and Sadoff $(2012)$ give four reasons why players might play continue in the normal version of this game where the sum of the payoffs increases exponentially in each stage of the game. List and explain each of these reasons, and explain whether each reason applies to this constant sum version of the game.

c$)$ Assume Player $1$ is rational and self$-$interested. Find the "behavioral game theory" Subgame Perfect Equilibrium if:

i$)$ Player $2$ is motivated by positive reciprocity. Her payoff is $\backslash ($ c$\_\{2\}+\backslash$alpha c$\_\{1\}\backslash )$ if player $1$ has played continue and she rewards her for doing so by playing continue. The weight that Player $2$ puts on Player $1\text{'}$s payoff is $\backslash (50\backslash \%\backslash )($so $\backslash (\backslash$alpha$=0\mathrm{.}5\backslash )).$

ii$)$ Player $2$ is motivated by inequity aversion. Her payoff is $\backslash ($ c$\_\{2\}-\backslash$underline$\{\backslash$lambda$\}\backslash$left$(\backslash$left$|$c$\_\{2\}-$c$\_\{1\}1\backslash$right$|\backslash$right$)\backslash ).($Note: $||$ means absolute value: the difference in payoffs is valued negatively no matter which player is at a disadvantage.$)$ The weight that Player $2$ puts on the difference between the payoffs is $\backslash (100\backslash \%\backslash )($so $\backslash (\backslash$lambda$=1\backslash )).$

When I ran a similar experiment at Bentley, most players played stop very early in the game. Given your answers, does this fact suggest that Bentley students are more motivated by altruism or inequity aversion? Explain.