The following text is from the book: Bargaining and Markets by Osborne and Rubinstein, Academic Press Inc.
Page 17 under the chapter The Axiomatic Approach: Nash's Solutions:.
Two individuals can divide a dollar in any way they wish. If they fail to agree on a division, the dollar is forfeited. The individuals may, if they wish, discard some of the dollar. In terms of our model, we have $A = \{(a_1, a_2) \in \mathbb R^2: a_1 + a_2 \le 1$ and $a_i > \ge 0$ for $i = 1, 2\}$ (all possible divisions of the dollar), and $D = (0, 0)$ (neither player receives any payoff in the event of disagreement). Each player is concerned only about the share of the dollar he receives: Player $i$ prefers $a \in A$ to $b \in A$ if and only if $a_i \gt b_i (i = 1, 2)$.
Thus, Player $i’s$ preferences over lotteries on $A$ can be represented by the expected value of a utility function $u_i$ with domain $[0, 1]$.
My question: In the last line what good does it serve by using "expected value" in the line:
expected value of the utility function $u_i$ with domain $[0,1]$
Instead, what will we lose if we just write: "...represented by the utility function $u_i$ with domain $[0, 1]$."$?$ (i.e we drop expected value)
EDIT 1:
(added after reading @Shane 's answer)
In the above question
Set $A = \{(a_1, a_2) \in \mathbb R^2: a_1 + a_2 \le 1$ and $a_i \ge 0$ for $i = 1, 2\}$, $D=(0,0) \in A$, $D$ is the disagreement event. For each Player $i$ there is a function $u_i:A \cup \{D\} \to \mathbb R$ , called a utility function, such that one lottery is preferred to another if and only if the expected utility of the first exceeds that of the second.
Here I am not able to visualize a lottery $L$ as is mentioned in the answer by @Shane. What will this lottery look like? When calculating the expected value, which all elements of $A$ will we take into consideration (since $A$ is a non denumerable infinite set)?
PS: It is taken from the famous, Nash's Axiomatic Bargaining Theory.
EDIT 2:
@Shane
I know about vNM utility theorem. The problem is, I am not able to apply it here. Here $A = \{(a_1, a_2) \in \mathbb R^2: a_1 + a_2 \le 1$ and $a_i \ge 0 \}$. So $A$ has infinite elements. A typical $A$ would look like $\{(0.1,0.9),(0.2,0.4),(0.5,0.1),(0,0)...\}$. It will have infinite elements. In this particular example let us call $p=(0.1,0.9),q=(0.2,0.4),r=(0.5,0.1),s=(0,0)$. So for Agent $1$, $r\succ q \succ p \succ s $ since $r_1\gt q_1 \gt p_1 \gt s_1$. (If choosing among the above four alternatives only).
Here, Nash says that
each player's preference be defined on a set of lotteries over possible agreements and not just on the set of agreements themselves.
Further:
one lottery is preferred to another if and only if the expected utility of the first exceeds that of the second.
Let us say that $u_i(a)=a_i$ where $a \in A$ and $i=1,2$
So if I try to construct a lottery ,
say $L=t_1 \cdot p+t_2\cdot q+t_3\cdot r+t_4\cdot s$ where $\Sigma t_i = 1$ and $M=x_1\cdot p+x_2\cdot q+x_3\cdot r+x_4\cdot s$ where $\Sigma x_i = 1$ and $p,q,r,s \in A$.
By vNM, for agent 1, $L \succ M$ iff $t_1u_1(p)+t_2u_1(q)+t_3u_1(r)+t_4u_1(s) \gt x_1u_1(p)+x_2u_1(q)+x_3u_1(r)+x_4u_1(s)$
OR, $t_1p_1+t_2q_1+t_3r_1+t_4s_1 \gt x_1p_1+x_2q_1+x_3r_1+x_4s_1$
Right?
My doubts:
here I just picked up $4$ elements of $A$. But there are infinite such elements in $A$. So how will I come up with a lottery? I mean we will not get such $t_i $ or $x_i $ such that $\Sigma t_i = 1$ or $\Sigma x_i = 1$.
EDIT 3
Our agreement set is as usual $A=\{a_1, a_2, a_3 \cdot \cdot \cdot\}$ where $a_k$s are various agreements over which the two agents can reach. If the two agents do not reach to an agreement then disagreement event $D=(d_1,d_2)$ is invoked. Let a general element of the set $A$ be denoted by $a$. For example in the case of our “splitting the dollar game” the agreement set, $A = \{(a_1, a_2) \in \mathbb R^2: a_1 + a_2 \le 1$ and $a_i \ge 0$ for $i = 1, 2\}$. Here $D=(0,0)$
Now the text (in the above mentioned book) says:
Denote by $p \cdot a$ the lottery in which the agreement $a \in A$ is reached with probability $ p \in [0, 1]$ and the disagreement event $D= (d_1,d_2)$ occurs with probability $1 – p$. Let $\succcurlyeq _i$ be Player $i’$s preference ordering over lotteries of the form $p \cdot a$, and let $\succ _i$ denote strict preference. Consider an agreement $a^*$ with the property that for $(i, j) = (1, 2)$ and $(i, j) = (2, 1)$, for every $a \in A$ and $p \in [0, 1]$ for which $p \cdot a \succ_i a^*$ we have $p \cdot a^* \succcurlyeq_j a$.
My doubt is how can we construct a lottery like $p \cdot a$ where sum of probabilities is not equal to 1. Should not it(the lottery) be something like $p \cdot a + (1-p) \cdot D$ ?
The key is that player $i$ has preferences over lotteries, not just the outcomes themselves. Technically speaking, we want to know not only $u(x)$ for $x \in [0,1]$, but also $u(L)$ where $L$ is a lottery. One example is that $L$ could be a lottery with a 50% probability on $0$ and a 50% probability on $1$. As for why they want to allow for lotteries here, I can't say as I don't have the textbook in front of me right now. But, generally speaking, we do that in game theory just to allow for the possibilities of mixed actions and mixed Nash equilibria. If this is still unclear, please post a comment below and I can try to amend my answer to clarify further.
By the way, I teach game theory, and I personally probably wouldn't take credit off if you didn't mention expected there on an exam, unless that was specifically the point of the question. So you're right to think it's a somewhat nuanced, and not necessarily that motivated, inclusion.