Here is how the experiment goes:
Half of the participants were given the role of a first mover, and half that of the second mover. In each round, it was common knowledge that a first mover would be randomly matched with a second mover. The rules and procedures were public information to participants. In each round, the first mover was given 10.00 dollars and s/he had to decide how much (if any) to keep and how much to pass to the second mover. What was passed would be tripled before being received by the second mover. The second mover then had to decide how much (if any) to keep and how much (if any) to pass back. The first mover earnings were equal to the amount kept initially plus the amount passed back. The second mover earnings were equal to the amount kept in the second stage. In addition to what was earned (if any) from the pass/keep process, both players received an endowment, i.e., the first mover received 10.00 dollars, and the second mover received 10.00 dollars.
The treatments :
The experiment consisted of two treatments. The first treatment was played between round 1 and 5, and the second between round 6 and 10. The treatments were identical except for the endowment received by the two players. In the first treatment, both players received no endowment. In the second treatment, the first mover received 10.00 dollars in each round (in addition to the amount that s/he was given to divide), and the second mover received $0.00.
My question is, what is the Nash equilibrium for this game assuming that individuals care only for maximizing their own earnings and that this is common knowledge amongst players?
My guess is that the Nash equilibrium is to defect in all cases as in the first mover should keep all the money he gets and so should the second mover. Is that correct? why?
It seems that you are right. First of all, the endowment doesn't matter since both players get it no matter what. Let's consider the game from the end. In round 10 after the second player receives (maybe) his payment, he won't give anything back because it is a clear loss for him. So the first player is not going to give him any money in the first place. Then we can make similar reasoning for all previous rounds inferring that they will be constantly defecting each other. More formally every other strategy cannot be an equilibrium because the player making the last nonzero transfer can always decide not to do so and make himself better off.