Question 2: Explain what is meant by the St Petersburg paradox. How does the model of choice under uncertainty explain this paradox? Are there any problems with the model of choice under uncertainty?
The St. Petersburg Paradox is a famous problem in decision theory and probability introduced by the Swiss mathematician Daniel Bernoulli in 1738. The paradox involves a game of chance with an infinite expected payout, yet most people would not be willing to pay more than a modest amount to play the game (Fontana, & Palffy-Muhoray, 2020). In other words, the game has a high potential payout but a low expected utility. The game begins with a coin toss, and if the coin lands heads, the player receives $2. If the coin lands tails, the game continues with another coin toss, and if the second toss lands head, the player gets $4 (Fontana, & Palffy-Muhoray, 2020). The game continues with each successive coin toss doubling the potential payout until the coin lands tails, at which point the game ends, and the player receives nothing. The expected payout for this game is infinite, as the probability of the coin landing tails on any given toss is 1/2, and the potential payout for each toss doubles each time. Despite the infinite expected payout, most people would not be willing to pay more than a modest amount to play the game. This apparent paradox has puzzled mathematicians and economists for centuries. In recent years, it has been explained using the model of choice under uncertainty, which assumes that people evaluate outcomes in terms of their expected value and perceived utility.
The St. Petersburg paradox involves a game with the following rules: a player pays a fixed amount to enter the game, and then a fair coin is flipped repeatedly until it lands on tails. The player receives $2^ {n} $ dollars, where n is the number of times the coin was flipped before landing on tails (Fontana, & Palffy-Muhoray, 2020). The paradox arises because the game's expected value is infinite, but many people would rather avoid paying a very high entry fee to play.
To calculate the expected value of the game, we can use the formula:
- E(X) = ∑ (i=1 to ∞) P(X=i) x i
Where, X is the random variable representing the player's winnings, and P(X=i) is the probability of winning i dollars. Since the probability of flipping tails on the first flip is 1/2, the probability of winning 2 dollars is also 1/2. Similarly, the probability of flipping tails on the second flip is 1/4, so the probability of winning 4 dollars is also 1/4. In general, the probability of winning 2^ {n} dollars is (1/2) ^ {n+1}. Using these probabilities, we can calculate the expected value of the game as follows:
- E(X) = ∑ (n=0 to ∞) (1/2) ^ {n+1} x 2^ {n}
Which simplifies to:
- E(X) = ∑ (n=0 to ∞) 1/2 = ∞
As we can see, the game's expected value is infinite, which means that, on average, a player can expect to win an unlimited amount of money if they play this game many times. However, many people are not willing to pay a very high entry fee to play, and this is where the paradox arises. The infinite expected value problem occurs because the formula for expected value assumes that individuals have infinite wealth and can therefore afford to pay any entry fee for the game. However, in reality, individuals have limited wealth, so their decision to play the game is affected by the cost of entry. As we will discuss further in the next section, the model of choice under uncertainty can help explain this paradox.
Expected utility theory is a model that describes how individuals make choices under uncertainty. According to this theory, individuals evaluate the potential outcomes of a decision and the probability of each outcome occurring (Szollosi et al 2019). The outcomes are then weighted by their utility or value, which is subjective and varies from person to person. The expected utility of a decision is calculated by multiplying the utility of each outcome by its probability of occurring and then summing these products. The von Neumann-Morgenstern axioms are a set of assumptions that ensure consistency in an individual's choices under uncertainty. These axioms require an individual's preferences over uncertain outcomes to satisfy transitivity, completeness, continuity, and independence. Independence means that an individual's preferences between two decisions should not depend on the outcomes of other decisions.
Expected utility theory resolves the St. Petersburg Paradox by considering the diminishing marginal utility of wealth. As the amount of wealth increases, the marginal utility, or additional satisfaction, from each additional wealth unit decreases (Szollosi et al 2019). Therefore, the expected utility of a decision involving a high potential payout is reduced because the utility gained from the additional wealth decreases as more wealth is accumulated. In other words, the utility gained from a large payout needs to be more significant to compensate for the low probability of winning it. This leads individuals to prefer a smaller payout with a higher probability of occurring rather than a more significant payout with a lower probability of occurring.
Expected utility theory has been the dominant decision-making model under uncertainty for many years, but it has its critics. One of the main criticisms of the theory is that real-world decision-makers can violate it. Specifically, some decision-makers appear to violate the von Neumann-Morgenstern axioms. For example, they may exhibit preferences that violate the transitivity axiom, meaning they may prefer option A to option B and option B to option C, but prefer option C to option A. This violates the principle of rationality and undermines the expected utility theory's ability to predict decision-making (Szollosi et al 2019). Experimental evidence suggests that individuals only sometimes behave according to the expected utility theory. For instance, the "reflection effect" demonstrates that individuals often exhibit risk aversion when faced with gains but risk-seeking behavior when faced with losses. This behavior contradicts the expected utility theory's assumption of risk neutrality. Additionally, the Allais paradox, similar to the St. Petersburg paradox, suggests that individuals often make decisions that violate the independence axiom.
As a result of these criticisms, alternative decision-making models under uncertainty have been proposed. These models seek to capture the cognitive and emotional biases affecting real-world decision-makers. For instance, prospect theory is a popular alternative to expected utility theory, accounting for the reflection effect and other biases. The theory introduces the concept of "loss aversion," which suggests that individuals are more sensitive to losses than gains. It also suggests that individuals are more likely to make decisions based on relative rather than absolute outcomes. Other alternative models include the rank-dependent utility theory, which accounts for the diminishing marginal utility of wealth, and the cumulative prospect theory, which accounts for how people weigh probabilities (Fontana, & Palffy-Muhoray, 2020). These alternative models are gaining traction in economics and other fields, and they represent an important step forward in our understanding of decision-making under uncertainty.
Question 5: "Majority voting results in chaotic decisions". Discuss this statement in the context of Arrow's theorem.
Voting is essential for making collective decisions. Majority voting is one of the most commonly used methods to reach a decision in a democratic society. However, the idea that majority voting always leads to an unambiguous decision is only sometimes true. In some cases, majority voting can result in chaotic outcomes, which could be unexpected and undesirable (Kloeckner, 2020). Arrow's theorem provides a framework to understand the limitations of majority voting and the conditions necessary for a fair and rational collective decision-making process.
Arrow's theorem highlights the inherent difficulties in aggregating individual preferences into a collective decision. The theorem shows that the properties of a voting system are in tension with each other, and it is impossible to satisfy all of them simultaneously. For instance, a voting system may be non-dictatorial but not Pareto efficient or Pareto efficient but not independent of irrelevant alternatives (Kloeckner, 2020). This means that any voting system will have limitations and flaws, and we need to accept that there is no perfect solution to the problem of collective decision-making.
Arrow's theorem is particularly relevant to the statement that "majority voting results in chaotic decisions." The theorem shows that the collective decision may not reflect the majority's preferences even in a simple scenario where individuals are asked to choose between two options. Moreover, Arrow's theorem suggests that this problem is not a result of a particular voting system but rather a fundamental limitation of any voting system. The potential for chaos in majority voting arises when individual preferences are not aligned (Kloeckner, 2020). This means that different individuals may have different preferences, and depending on the issue, these groups may overlap. In such situations, it is possible that no option has majority support or that the outcome of the vote changes when an irrelevant alternative is added or removed. This is precisely the kind of chaos that Arrow's theorem warns us about.
Arrow's theorem has significant implications for voting systems, widely studied in political science, economics, and social psychology. One of the most important implications of Arrow's theorem is that no perfect voting system satisfies all desirable properties. This means that any voting system will inevitably have limitations and drawbacks. As a result, the design of voting systems requires a trade-off between the desired properties. In other words, having a voting system that is non-dictatorial, Pareto efficient, and independent of irrelevant alternatives is impossible.
Another implication of Arrow's theorem is that majority voting is only sometimes fair and rational for making collective decisions. This is because majority voting violates the independence of irrelevant alternative property (Duddy, & Piggins, 2020). The independence of irrelevant alternatives property states that adding or removing an option that is not chosen should not change the outcome of the vote. In other words, choosing an irrelevant option should not affect the ranking of the relevant options. However, majority voting can be vulnerable to manipulation by adding irrelevant options that can change the vote outcome. For example, in an election with two candidates, adding a third, irrelevant candidate can split the vote and result in an outcome different from what the voters would have chosen without the irrelevant candidate.
Moreover, majority voting can result in chaotic outcomes when there are three or more options. In this case, there is a possibility that no option will receive a majority of votes, and there will be no clear winner. This situation is known as a cycling paradox or a Condorcet paradox. A Condorcet paradox occurs when the pairwise majority preferences between three or more options are inconsistent (Duddy, & Piggins, 2020). For example, suppose there are three options: A, B, and C. Suppose that in the first round of voting, A receives the most votes. In the second round of voting, B receives the most votes, and C receives the most in the third round. There is no clear winner in this case, and the vote outcome could be more manageable.
The Condorcet paradox illustrates how majority voting can lead to chaotic outcomes and highlights the limitations of using majority voting as a method for making collective decisions. However, Arrow's theorem also suggests no perfect alternative to majority voting. Arrow's theorem proves that any voting system that satisfies certain reasonable assumptions will inevitably have limitations and drawbacks. For example, one alternative to majority voting is ranked-choice voting, where voters rank the options in order of preference (Duddy, & Piggins, 2020). However, ranked-choice voting can also lead to cycling paradoxes and violates the independence of irrelevant alternative property.
Arrow's theorem is a fundamental result in a social choice theory that shows the impossibility of designing a voting system that satisfies desirable properties. This theorem has several implications for voting systems, including that no perfect voting system satisfies all desirable properties and that majority voting is not always fair and rational for making collective decisions. The Condorcet paradox illustrates how majority voting can lead to chaotic outcomes, but Arrow's theorem also suggests that there is no perfect alternative to majority voting (Duddy, & Piggins, 2020). As a result, the design of voting systems requires careful consideration of the trade-offs between different desirable properties. There is a need for ongoing research on designing fair and rational methods for making collective decisions.
Consequently, majority voting is a widely used method for making collective decisions. However, as Arrow's theorem shows, no perfect voting system satisfies all desirable properties. Majority voting can result in chaotic outcomes when there are three or more options (Kloeckner, 2020). The outcome of the vote can be unpredictable, and there may be no clear winner. Therefore, it is important to understand the limitations of majority voting and to design voting systems that are fair, rational, and efficient.
References
Duddy, C., & Piggins, A. (2020). Arrow’s Theorem. In
Oxford Research Encyclopedia of Politics.
Fontana, J., & Palffy-Muhoray, P. (2020). St. Petersburg paradox and failure probability.
Physical Review Letters,
124(24), 245501.
Kloeckner, B. R. (2020).
Bad cycles and chaos in iterative approval voting. ArXiV working paper.
Szollosi, A., Liang, G., Konstantinidis, E., Donkin, C., & Newell, B. R. (2019). Simultaneous underweighting and overestimation of rare events: Unpacking a paradox.
Journal of Experimental Psychology: General,
148(12), 2207.
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