"Is there any point to which you would wish to draw my attention?"
"To the curious incident of the dog in the night-time."
"The dog did nothing in the night-time."
"That was the curious incident," remarked Sherlock Holmes.
Source: Doyle, A. C. (1894). The Memoirs of Sherlock Holmes. London, England: George Newnes.
"Reports that say that something hasn't happened are always interesting to me, because as we know, there are known knowns; there are things we know we know. We also know there are known unknowns; that is to say we know there are some things we do not know. But there are also unknown unknowns — the ones we don't know we don't know. And if one looks throughout the history of our country and other free countries, it is the latter category that tend to be the difficult ones."
Donald Rumsfeld, February 12, 2002.
Albert and Bernard just met Cheryl.
"When's your birthday?" -- Albert asked Cheryl. Cheryl thought a second and said, "I'm not going to tell you, but I'll give you some clues." She wrote down a list of 10 dates:
May 15, May 16, May 19
June 17, June 18
July 14, July 16
August 14, August 15, August 17
"My birthday is one of these," -- she said.
Then Cheryl whispered in Albert's ear the month and only the month of her birthday. To Bernard, she whispered the day, and only the day.
"Can you figure it out now?" -- she asked Albert.
Albert: I don't know when your birthday is, but I know Bernard doesn't know, either.
Bernard: I didn't know originally, but now I do.
Albert: Well, now I know, too!
When is Cheryl's birthday?
"whispered in Albert's ear the month"
"To Bernard, she whispered the day"
Albert: I don't know when your birthday is,
but I know Bernard doesn't know, either.
Bernard: I didn't know originally, but now I do.
Albert: Well, now I know, too!
Let's consider an economy with three periods $t=0,1$ and $2$; and two sources of uncertainty: inflation (low or high) and unemployment (low or high). Therefore there are four states of the world:
The set of the states of the world is $$\Omega = \{ \omega_1, \omega_2, \omega_3, \omega_4 \}.$$ Usually, in an economy uncertainty is gradually resolved over time.
In an economy, uncertainty is resolved over time as new information is revealed over time. For example:
The knowledge of the agents is described by the following partitions:
The partitions reflect the gradual information revelation:
A filtration is a sequence of partitions $\mathcal{F} = (\mathcal{P}_0, \mathcal{P}_1, \ldots, \mathcal{P}_T)$ where each partition represents the information available at time $t$, and the partitions become progressively finer over time.
Formally, a filtration satisfies:
What does "finer" mean?
Partition $\mathcal{P}_s$ is finer than $\mathcal{P}_t$ if every element of $\mathcal{P}_s$ is a subset of some element of $\mathcal{P}_t$.
Intuition: As time progresses, agents can distinguish between more states. They never "forget" information—they only gain more precision about which state occurred.
In our inflation-unemployment example: $$\mathcal{F} = (\mathcal{P}_0, \mathcal{P}_1, \mathcal{P}_2)$$ where: $$\begin{aligned} \mathcal{P}_0 &= \{\{\omega_1, \omega_2, \omega_3, \omega_4\}\} \\ \mathcal{P}_1 &= \{\{\omega_1, \omega_2\}, \{\omega_3, \omega_4\}\} \\ \mathcal{P}_2 &= \{\{\omega_1\}, \{\omega_2\}, \{\omega_3\}, \{\omega_4\}\} \end{aligned}$$
Notice that:
This confirms $\mathcal{F}$ is indeed a filtration.
The lottery that gives prize $x$ with probability one (with certainty) is denoted by: $$\delta_x=\left((x),(1)\right).$$
The expected value of a lottery $\ell_1=\left((x_1,x_2,\ldots,x_n),(p_1,p_2,\ldots,p_n)\right)$ is: $$\mathbb{E}[\ell_1]=p_1\cdot x_1+p_2\cdot x_2 +\ldots p_n\cdot x_n = \sum_{i=1}^n p_i\cdot x_i;$$ and the variance of a lottery $\ell$ is: $$\begin{aligned}\mathrm{Var}[\ell_1]=&p_1\cdot (x_1-\mathbb{E}[\ell_1])^2+p_2\cdot (x_2-\mathbb{E}[\ell_1])^2 +\ldots p_n\cdot (x_n-\mathbb{E}[\ell_1])^2 = \\ =& \sum_{i=1}^n p_i\cdot (x_i-\mathbb{E}[\ell_1])^2.\end{aligned}$$
Consider two lotteries $A=(x_0,1)$ and $B=\left((x_0+M,x_0-M),\left(\frac{1}{2},\frac{1}{2}\right)\right)$ with $M>0$. That is, $A$ is a sure lottery (no risk) and $B$ is a risky lottery in which you win or lose $M$ with equal probabilities.
Notice both lotteries have the identical expected value, $\E[A]=\E[B]=x_0$.
The difference in expected utility of these lotteries is approximately,
$$U(B)-U(A)=\frac{M}{2}\cdot\left(Mu(x_0+M)-Mu(x_0-M)\right)$$Decision Problem: Before leaving home, choose whether to carry an umbrella, a parasol, or nothing.
Set-up:
Strategies: A strategy assigns an action to each information set.
Strategy $s^*$: parasol if sunny, umbrella if not sunny.
Utility matrix $u(\text{action}, \omega)$:
| $\omega_1$ (sunny) | $\omega_2$ (cloudy) | $\omega_3$ (rainy) | |
|---|---|---|---|
| umbrella | 6 | 7 | 10 |
| parasol | 12 | 6 | 2 |
| nothing | 10 | 8 | 0 |
Expected utility of strategy $s^*$ (parasol if sunny, umbrella if not):
$$\begin{aligned} \mathbb{E}[u|s^*] &= \Pr(I_1) \cdot u(s^*(I_1), \omega_1) + \Pr(I_2) \cdot \mathbb{E}[u|I_2, s^*] \\ &= \frac{1}{2} \cdot u(\text{parasol}, \omega_1) + \frac{1}{2} \cdot \left[\Pr(\omega_2|I_2) \cdot u(\text{umb}, \omega_2) + \Pr(\omega_3|I_2) \cdot u(\text{umb}, \omega_3)\right] \\ &= \frac{1}{2} \cdot 12 + \frac{1}{2} \cdot \left[\frac{1}{2} \cdot 7 + \frac{1}{2} \cdot 10\right] = 6 + \frac{17}{4} = 10.25 \end{aligned}$$Decision Problem: The Federal Reserve must decide whether to lower interest rates or keep them unchanged.
Set-up:
Strategies: A strategy assigns an action to each information set.
Strategy $s^*$: lower if low inflation, keep if high inflation.
Utility matrix $u(\text{action}, \omega)$:
| $\omega_1$ (LI, LU) | $\omega_2$ (LI, HU) | $\omega_3$ (HI, LU) | $\omega_4$ (HI, HU) | |
|---|---|---|---|---|
| lower | 6 | 10 | 2 | 4 |
| keep | 8 | 5 | 9 | 7 |
Expected utility of strategy $s^*$ (lower if low inflation, keep if high):
$$\begin{aligned} \mathbb{E}[u|s^*] &= \Pr(I_1) \cdot \mathbb{E}[u|I_1, s^*] + \Pr(I_2) \cdot \mathbb{E}[u|I_2, s^*] \\[4pt] \mathbb{E}[u|I_1, s^*] &= \Pr(\omega_1|I_1) \cdot u(\text{lower}, \omega_1) + \Pr(\omega_2|I_1) \cdot u(\text{lower}, \omega_2) \\[4pt] &= 0.8 \cdot 6 + 0.2 \cdot 10 = 6.8 \\[4pt] \mathbb{E}[u|I_2, s^*] &= \Pr(\omega_3|I_2) \cdot u(\text{keep}, \omega_3) + \Pr(\omega_4|I_2) \cdot u(\text{keep}, \omega_4) \\[4pt] &= 0.6 \cdot 9 + 0.4 \cdot 7 = 8.2 \\[4pt] \mathbb{E}[u|s^*] &= 0.5 \cdot 6.8 + 0.5 \cdot 8.2 = 7.5 \end{aligned}$$The utility of wealth drawn on the back of the block has decreasing marginal utility below $x=100$ and increasing marginal utility above it: $$u(x)=\begin{cases}10000-(100-x)^{2} & \text{if } x\le 100\\[2pt] 10000+(x-100)^{2} & \text{if } x\ge 100\end{cases}$$
Lottery A: a gamble over low wealth levels
$$A=\begin{cases}0 & \text{with probability } 1/2\\ 100 & \text{with probability } 1/2\end{cases}\qquad E_A=\tfrac{1}{2}\cdot 0+\tfrac{1}{2}\cdot 100=50$$
$$\begin{aligned}U(A)&=\tfrac{1}{2}\,u(0)+\tfrac{1}{2}\,u(100)=\tfrac{1}{2}\cdot 0+\tfrac{1}{2}\cdot 10000=5000\\[2pt] u(E_A)&=u(50)=7500\end{aligned}$$ So $U(A)\lt u(E_A)$: the chord lies below the curve. Over this range the individual is risk-averse and would pay to replace the lottery by its expected value.
Lottery B: a gamble over high wealth levels
$$B=\begin{cases}100 & \text{with probability } 1/2\\ 200 & \text{with probability } 1/2\end{cases}\qquad E_B=\tfrac{1}{2}\cdot 100+\tfrac{1}{2}\cdot 200=150$$
$$\begin{aligned}U(B)&=\tfrac{1}{2}\,u(100)+\tfrac{1}{2}\,u(200)=\tfrac{1}{2}\cdot 10000+\tfrac{1}{2}\cdot 20000=15000\\[2pt] u(E_B)&=u(150)=12500\end{aligned}$$ So $U(B)\gt u(E_B)$: the chord lies above the curve. Over this range the same individual is risk-loving and would pay to take the lottery instead of its expected value.