Often the payoff of a game is thought of as equal to the expected value of return from the game. We start with an example which shows why expected value of a game cannot be a measure of our confidence in the payoff from the game.
Suppose you are given an option to take Rs. 10 straightaway or Rs. 70 if a coin toss yields head (on tails you get Rs. 0). Then most of you will opt for the second option. Instead of this, if the options are Rs. 10 million straightaway or Rs.70 million on a coin toss, then there is little chance that you will opt for the second option, even though the expected return in the second case is more.
We see a similar situation that occurs in the casino in the next section.
gets a return of Rs. 2 if first coin toss gives heads, Rs. 4 if the first coin toss reveals tails and the second reveals heads, and so on as shown in the table above. In this example surely any gambler will be willing to invest Rs. 2 (i.e ) since he is sure to get atleast that much in return. Some gamblers might be willing to invest Rs. 4 but it seems unlikely that anyone will be willing to invest Rs. 1000.
But the expected return of this game is
Even though the expected return from the game is infinity few gamblers will be willing to invest an amount of Rs. 1000 since the probability of winning a good amount of money is very small and the risk involved is very high.
This example shows that the expected return is clearly not what is needed to be maximized for increasing confidence of a player in a game. Then what is to be maximized and why?
We try to model the preferences of a player of a game by a relation in the next section.
Recall that a game consists of:
Define a preference relation on the set of outcomes, , as follows. . The relation satisfies the following properties:
Totality implies that there is an ordering between any two outcomes. For any two outcomes atleast one of them is preferred as much as the other i.e. every two outcomes are comparable.
Reflexivity is due to the fact that an instance of an outcome is preferred as much as any other instance of the same outcome.
Suppose an outcome is preferred as much as outcome and is preferred as much as outcome . Then is preferred as much as . This is Transitivity.
A preference relation is rational when it is total, reflexive and transitive. We will be using rational preference relation from now on.
is said to be a utility function representing a preference relation if
If a player is indifferent to two outcomes, then the utlilites of the outcomes are same. The converse is also true.
For example, if the amount paid by the gambler is and the amount gained by the gambler is , then the utility function could be .
It is easy to see that if is a monotonically increasing function and the utility function represents then also represents . So also represents whenever .
Note that we cannot replace the phrase ``monotonically increasing'' by ``monotonically non-decreasing'' since the condition in equation above will not be satisfied. For example, when the utility function is constant, an outcome strictly preferred over ( but ) will satisfy and hence . But this contradicts .
Hence, in particular we cannot have in the earlier example of .
Define if & . Clearly for all representing .
If then we say that the player is indifferent to the two outcomes. She prefers each outcome as much as the other.
Lets consider now a game that will formalize our notion of value that we attach to a commodity.
The set of outcomes is a set of tuples
Let the utility function represent the preference relation of trader .
Let's also assume free disposal i.e. the traders can throw away any commodity without incurring any loss to themselves. Equivalently, having more of a commodity cannot decrease the total utility for a trader.
Also assume that the th commodity has no a priori utility. That is,
To start, each trader borrows certain amount of commodity from a centralised bank and each has to return the same amount to the bank after the trade is over.
Lets look at a particular trader . Let
be her utility function. Suppose she gets at a rate and is able to sell at rate . Then she would get amount of by spending amount of and sell amount of to get amount of back. If there is no change in her possession of (this is also called conservation of paper),
A trade involving her is possible only if she does not lose in utility. That is,
Assuming are continuous and differentiable, and the amount of trade involved is small, the right hand side is same as
Then the condition for trade is
which is same as
using the equation 5 above due to conservation of paper and the fact that .
If the above inequality is in the opposite direction then the trade is possible in the opposite direction (assuming the trader has a non-zero amount of the commodity).
The condition for no trade is no change in utility for small trades i.e.
In general for all commodities
When and , trade can happen only in the direction in which trader increases the amount of . Hence in this case equation 9 (derived assuming the same direction of trade) is the condition of equilibrium.
Here is the relative price of commodity . If the similar ratios of all traders are the same, there is no change of utility by any trade and this is a state of equilibrium.
But if the above ratios are not same for two traders, that is, if
with , trade can occur. For example let . Then trade can occur in the following manner:
Let . Trader can sell an amount of commodity to trader and buy an amount of from the same trader. From the point of view of trader , her change in utility is given by,
and from the point of view of trader , change in utility after this trade is,
Hence trade can occur and both traders end up increasing their utilities.
It is appreciable that with so few assumptions (free disposal, continuity and differentiability of utility functions, and a rational preference relation) we were able to show that the prices will be proportional to the ratio of change of utility and the change of quantity.
The above arguments reinforces the notion that value of a commodity is proportional to the ratio of change of utility and the change in quantity that forces that change in utility i.e.
This explains the high cost and low utility of gold compared with low cost and high utility of water, as shown in the figure. Since the marginal utility of water is much smaller than that of gold in the present state, the slope of the graph for gold is much more than for water.