Life

We have seen one type of ultimatum game where the player who receives the offer (Annie) has another option outside the one in the game. Those options are a weak type (0.25) or a strong type (0.5) and depend on a probability p in favour of the former.

Now, imagine Becky knows Annie’s outside option (0.25 vs 0.5). In that case, Becky will make 0.25 p times and 0.5 (1-p) times to get her offer accepted.

If p < 2/3

Becky will offer 0.25 (2/3)^rd of the time and 0.5 (1/3)^rd of the time. Her expected value of surplus in the case of complete information is
p x (0.75) + (1-p) x (0.5)
Note that in the case of incomplete information, she would have offered 0.5 all the time, leading to the expected value
p x (0.5) + (1-p) x 0
The difference between the two [p x (0.75) + (1-p) x (0.5) – p x (0.5)] is 0.25p. It is the value of information.

If p > 2/3

In the case of complete information, Becky will offer 0.5 (2/3)^rd of the time and 0.25 (1/3)^rd of the time. Her expected value of surplus is
p x (0.75) + (1-p) x (0.5)
Note that in the case of incomplete information, she would have offered 0.25 all the time, leading to the expected value
p x (0.75) + (1-p) x 0
The difference between the two [p x (0.75) + (1-p) x (0.5) – p x (0.75)] is 0.5(1-p). It is the value of information.

Bargaining 101 (#24): William Spaniel

The ultimatum game continues between Annie and Becky. But there is a difference here.

Like before, Becky is offering Annie a 0 to 1 division of the surplus, which Annie can accept or reject. However, instead of ending up with zero upon rejection, she has a payoff of 0.25 or 0.5 from an outside option. Unlike in earlier cases, Becky is uncertain about Annie’s bottom line. So, she attaches a probability p for 0.25 and (1-p) for 0.5. It means:

If Becky offers < 0.25, Annie rejects 100%
If Becky offers >/= 0.25 but < 0.5, Annie accepts with a probability p
If Becky offers >/= 0.5, Annie accepts 100%

If Becky offers 0.5, then her expected payoff is:
1 – 0.5 = 0.5

If Becky offers 0.25, then her expected payoff is:
p x (1-0.25) + (1-p) x 0 = 0.75 p

In other words, Becky should offer 0.25 if 0.75 p > 0.5 or p > 0.5/0.75 or p > 2/3

Bargaining 101 (#21): William Spaniel

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

We have seen the first level of bargaining – the ultimatum game. Becky made the offer for Annie’s car, and the latter will accept or reject it. The decision tree that we have seen before can be drawn as a fraction of the surplus:

In the next level, Annie can make a counteroffer instead of rejecting the offer. Here is the game:

The above picture suggests that Annie offers Becky a fraction of the surplus in her counteroffer. But there is a small problem. Since the counteroffer and the decision consume more time than the take-it-or-leave-it situation, the benefit from the subsequent game must be discounted (time value of money) by a fraction d.

Let’s solve the problem, as before, from the end backwards.

The solution is exactly like before, and the answer is that Annie will get the entire surplus, and Becky has no bargaining power. Here is the whole picture.

And to the start of the tree.

Now, Becky must decide what X should be, as she knows Annie can reject the offer and get a payoff of d. In that case, Becky will get zero. The solution that Annie accepts, therefore, is X = d. Annie is getting d, and Becky will get (1 – d) of the surplus.