Chapter 10: Risk Aversion

Question 1

Unsurprisingly, many of the decision made by Decision Makers are made with the goal of being _____________ _______________.

Answer 1

risk averse.

Question 2

Types of Risk Aversion (3)

Answer 2

1) Actuarial Risk Aversion
2) Aversion Against Utility Risks
3) Epistemic Risk Aversion

Question 3

Actuarial Risk Aversion
(Review Golf Ball Example in Notes)

Answer 3

A decision maker is risk averted if one, starts with certainty, is unwilling to bet in an actuarilly fair gamble.

Question 4

_____________ ____________ _______________ is consistent with _______________ ____________ ______________, and can be _____________ on a ______________.

Answer 4

Actuarial Risk Aversion; Maximizing Expected Utility; plotted; graph

Question 5

If the Decision maker’s __________ function for an object ($1M, apple, etc.) is “____________,” on a graph, then the more ___________ ______________ the decision maker is: the expected _______________ of a _____________ prize (___) is greater than the ____________ of ______________ (___ ___) and _____________ prizes (___).

(Review Apple Gamble Example in Notes)

Answer 5

utility; concave; risk averse; utility; small; (n); utility; larger; (2n); smaller; (n)

Question 6

Dotted Linear Line on Graph means…

(Review Apple Gamble Example in Notes)

Answer 6

Represents someone who is actuarilly risk neutral: the expected utility of 2n and 0 are the same.

Question 7

Solid “Concave” Line on Graph means…

(Review Apple Gamble Example in Notes)

Answer 7

Represents someone who is actuarilly risk averse: utility of n is greater than 2n and 0.

Question 8

Aversion to Utility Risks (Rabin Explanation)

Answer 8

• It’s a Maximizing Expected Utility Orthodoxy (doesn’t use / inconsistent with MEU)
• Takes issue with Allais Paradox because the requirements imposed by the Allais Paradox are preposterous.
• Ex. You decline a gamble wherein winning will earn you $110 and losing will cost you $100. Following from the MEU framework, this means you would also necessarily have to decline a gamble wherein losing cost you $600, and where you could win whatever amount of money whatsoever.
• A step further, let’s say you decline a gamble between winning $125, and losing $110. According to the MEU framework, there’s NO amount of money WHATSOEVER that could be offered to you, in winning, in a gamble in which you lose $1,000.
• These obligations are ludicrous. An agent’s marginal utility for money deteriorates far too fast, caused ABSURD RISK AVERSION. it cost the agent potentially lucrative gambles, that would be rationally permissible to accept.
• These obligations, based on an agent’s preferences for their current level of wealth over a gamble between some amount of money above and below their current level of wealth, holds as long as one’s utility function is concave.

Question 9

Aversion to Utility Risk (Rabin Theorem)

Answer 9

u (W + z) = (10/11)ʸ * u(x) ; where y = 1 + ((z - 11)/21)

W: Current level of wealth
z: value above your current level of wealth (addition above current level of wealth)
10: potential loss
11: potential gain
y: Number of Dollar Windows
u(x): Utility of W - potential loss (10)
21: potential loss (10) + potential gain (11)

Question 10

Risk Aversion does have an ________________ component, and is ________________ with the principle of _________________ _________________ ___________________, and therefore also inconsistent with _________________ risk aversion.

Answer 10

Epistemic; inconsistent; maximizing expected utility; Actuarial

Question 11

Explain the Tennis Match

(Cross-references notes as needed)

Answer 11

You attend a set of 3 Tennis Matches: A, B, C

Match A:
- You know each player very well. Based ont their respective skill sets, you believe they’re evenly matched, and hence assign a 50/50 chance to each player for winning
- You honestly believe each player has an equal probability of winning.
- Here, your assignment of probability assessments is epistemically robust.

Match B:
- You don’t know either of the players very well, or their skill sets.
- So you have no choice but to be an equal opportunist, and assign each player a probability of 50/50 of winning the match.
- Here, your assignment of probability assessements are NOT epistemically robust.

Match C:
- Similarly to Match B, you don’t really know each player at all / skill sets.
- All you know is that these players aren’t evenly matched; one player is far better than another, HOWEVER, you don’t know who this player is among the two.
- So, you have no choice but to assign a 50/50 probability to each of the player’s winning here as well.

You are now offered to place bets on each event. You will either win or lose $100 each match depending on who wins the match.

Question 12

Gardenfors and Salhin, the Philosophers who constructed this tennis match problem state that…

1) If one’s willing to place a bet of _________ odds in the ____st Match (__), then they should necessarily be willing to place equal odds for the other _______ matches (__ and __).

2) HOWEVER, it’s rational that one would only place a _______ on the ________ Match and not the others, GIVEN that they have _____________ KNOWN probabilities for the _____st Match only.

Answer 12

equal; 1st; A; two; B; C

bets; 1st; RELIABLE; 1st

Question 13

This Tennis Match Problem is meant to demonstrate how one is more comfortable with making decisions based on ______________ probabilities, and are _____________ more preferrable than making decisions based on ______________ probabilities.

Answer 13

known; ALWAYS; unknown

Question 14

How would you assign probability functions to Match C ?

(Refer to notes as needed)

Answer 14

• Keep in mind, you don’t really know anything about the players, other than that one of the players is much better than the other.

So, the best we can do is assign TWO separate probability functions: where one of the players has a higher probability of winning and the other is not, then vice versa.

Probability Function 1) Player One = 0.9, Player Two = 0.1
Probability Function 2) Player One = 0.1, Player Two = 0.9

Question 15

How would you assign proability functions in Match B?

(Refer to notes as needed)

Answer 15

• Well here we don’t know anything about our respective players in this match.
• We don’t know any of the probability functions that could be applied to either of the players [that make up 1.0], so we would have to assign several probability functions to each player’s winning the game.
• …Such that the number of subjective probabilities we’d have to assign to each player winning the game is infinite.

Question 16

How would you assign probability functions in Match A?

(Refer to notes as needed)

Answer 16

• This should be relatively simple. Here, we know everything about the players and believe there’s an equal probability that either player could win the match.
• We assign a subjective probability of 0.5 (50/50) to each player.

Question 17

Considering Epistemic Risk Aversion is inconsistent with Actuarial Risk Aversion, and seeks not to use the Principle of Maximizing Expected Utility, what rule are we going to use to exercise risk aversion, and assess whether or not we should bet on any of the Tennis Matches (or any kind of event)?

Answer 17

• The Principle of Maximin Criterion of Expected Utility (MMEU).

Question 18

Maximin Criterion of Expected Utility (MMEU)

Answer 18

Says decision makers should (act in a way that would) maximize the minimal level of expected utility.

Question 19

Rules for Tennis Match Bets (ONE-SIDED CARD)
- Winning $100 on a bet = +10 units of utility.
- Losing $100 on a bet = -9 units of utility lost.

Minimal Expected Utility for Refusing a Bet = 0

Question 20

Match A: Should you bet or not? Answer yes or no, and show your work.

Answer 20

We established earlier the probability of each player winning is 0.5.

(10 × 0.5) + (-9 × 0.5) = 0.5

Okay well, 0.5 > 0 (the Minimal Expected Utility for Refusing the bet), so YES one should bet on Match A.

Question 21

Match C: Should you bet or not? Answer yes or no, and show your work.

Answer 21

Remember, we said there are two probability functions for Match C:
- Player 1 = 0.9 & Player 2 = 0.1
- Player 1 = 0.1 & Player 2 = 0.9

(10 × 0.9) + (-9 × 0.1) = 8.1
(10 × 0.1) + (-9 × 0.9) = -7.1

The first probability suggest we should place a bet on Match C (8.1 > 0), yet the second probability function suggest otherwise (-7.1 < 0). SO OVERALL, YOU SHOULD NOT PLACE A BET ON MATCH C.

Question 22

Match B: Should you bet or not? Answer yes or no, and show your work.

Answer 22

We don’t anything about the players in this match, and established that we’d have to assign an infinite number of probability functions to each player’s chances of winning the match.

• So, more straightforwardly and for more convenient reasons, we should simply collapse Maximin Criterion for Expected Utility into simply the Maximin Rule (choose the best worst outcome).
• Obviously the worst outcome would be to lose a $100 (take a loss of 9 utility units). So we’ll apply a probability of 0 to winning a $100 (+10) on a bet, and 1 to losing $100 (-9). So our probability function will look like…

(10 × 0) + (-9 × 1) = -9
Well -9 is surely less than 0 (the minimal expected utility for refusing a debt), so we should NOT take the bet.