We have seen a single-stage – ultimatum – game and a two-stage game.
Ultimatum game
In the single-stage game, Becky makes the offer. As the game is over after stage 1, the earnings are:
Annie: 0
Becky: 1
Two-stage game
In the two-stage game, Becky calls the shots. In the end, as we have seen, Annie earns d and Becky 1-d.
Annie: d
Becky: 1-d
Three-stage game
What happens if there is a stage 3? This happened because Annie had rejected Becky’s offer in the second stage. Now, Becky is making the counteroffer. If Becky’s offer is rejected, Annie will get nothing, and Becky will take the entire surplus of 1.
Annie: 0
Becky: 1
Annie knows in the second stage itself that Becky will get the full value (discounted by d due to the time delay) if the game goes to the third. So, in the second stage, Annie must offer d to get Becky to accept the offer. Annie can take the remaining 1-d.
Annie: 1-d
Becky: d
Knowing what returns Annie will get in the second stage (1-d), Becky can offer d(1-d) from the beginning itself for Annie to accept.
Annie: d(1-d) = d – d2
Becky: 1 – d(1-d) = 1 – d + d2
Summary
| Game | Offer by | Return Annie | Return Becky |
| 1-stage | Becky | 0 | 1 |
| 2-stage | Annie | d | 1 – d |
| 3-stage | Becky | d – d2 | 1 – d + d2 |
Extrapolating from the above pattern, we can say if the game was ending in stage 4, Annie would have made an offer and earned d – d2 + d3 and Becky would have ended up with 1 – d + d2 – d3
