Prerequisites: Chapters 1, 3, 4, and Dividing Six Items II.
This chapter is a continuation of the page "Dividing Few Items II", where games are analyzed and compared that distribute few items to two players, where each player may value different items differently. In this chapter we concentrate on what happens if the players know their value they assign to the items, but not their opponent's values. Such games are called games of incomplete information. These variants for the games ABABAB, ABBABA, ABBAAB, CUT&CHOOSE, RANDOM&EXCHANGE2 are briefly discussed in this chapter. See the previous chapter for description of these games.
Even in such games of incomplete information, the two players will have some beliefs about the values the other sister attaches. These beliefs could be based on rumors or hearsay. But such initial beliefs could also change during the play, since they could be obtained through observations of the decisions of the other player during the game. We will mainly assume here that the players believe that the values of their sister are not related to their own values, in other words, that the preferences of the two players are independent. Other possible beliefs are that the preferences are parallel, or opposed.
With these beliefs of independent preferences, it is rather obvious how the player play the three games--- ABABAB, ABBABA; and ABBAAB--- of dividing the items one-by-one: They would just play what we called "greedy strategy" in the previous chapter---each player chooses the item avaliable having the largest value to her. There is no reason to play tactical and choose something of lesser value, since no prediction is possible about what the other player will choose next. More precisely, all items could be chosen with equal probability.
As in the previous chapter, I ran 2000 simulations for random and independent values
of the items from 0 to 1, with uniform probability distribution.
The outcomes with respect to fairness and efficiency are displayed in the graph
to the right. The red dots are the incomplete information cases, and the green dots those
with complete information already mention before. Interestingly the outcomes of the games
ABABAB, ABBABA, and ABBAAB are more efficient in the incomplete information case,
whereas the fairness does not suffer much. Thus even in the complete information case,
it would be beneficial for both players if both would stick to the greedy strategy.
The problem is the same as with the Prisoner's Dilemma---they would not stick to it,
but rather deviate at some point. This result could also be expressed by saying
that complete information about the other players may be not benefitial to the players,
at least under certain conditions.
modified fairness ...................
CUT & CHOOSE: Different to the games from the "Dividing One-by-one" family, CUT AND CHOOSE is much more difficult to analyze with incomplete information than with complete information.
Note that the MAXIMIN strategy does not depend on the opponent's payoffs. Therefore this strategy could alsways be played, even in games of incomplete information. In the CUT & CHOOSE example, MAXIMIN means that Ann cuts the six items into two heaps whose values for Ann are as close together, as close to 50%, as possible. The MAXIMIN value would then be the minimum of these values.
However, under the same assumptions on beliefs of the players, Ann will actually play a little different. Assume as an example that items D and E have each a value of 3 for Ann, and that items F, G, H, and I each have a value of 1 for Ann. In the MAXIMIN spirit, Ann would pessimistically assume that Beth woud choose heap D-E, and leave her with a relative utility of 4/10=40%. However, isn't it much more likely that Beth would prefer the four item heap F-G-H-I than the two item heap D-E, in particular when we believe that values of the items for Anna dn beth are independent, and that we cannot conclude from the high value of items D and E for Ann that these items are also more valuable for Beth than the other four. ................
What we need to finish our analysis are probabilities that a two-item heap is more valuable than a four-item heap, or that a one-item heap is more valuable than a five-item heap. This probability could be obtained using simulation again (see ...), but it could also be computed as follows: ...............
How would the players perform the exchanges when they don't know each other's values? Since Ann has no information at all, the best she can do is to ask for the item of Beth that has the highest value for Ann. Of course, she could "bluff" and ask for another one, but for what reason? Beth would do the same...
The situation is slightly different for the second exchange, since now both players have a little information. However, this information is not really useful .... Therefore ...
Still, the results of CUT & CHOOSE and RANDOM & EXCHANGE 2 in simulations are far away from those of the dividing one-by-one games. No information on the values of the opponent affects Ann's ability to choose a good division into heaps a lot. The same with RANDOM & EXHANGE 2.