1 UMID: 30412153 Word Count: 1304 PHIL443 Package Principle Package principle assumes that the value the agent attached to the collection of bets is the sum of the values the agent attached to the bets individually (Hájek). This assumption considers the agents in the model value money or payoffs of the bets linearly. Also, the package principle demands the agents in the model regarding the placement of one bet in a package is not correlated with the outcome of another bet in the package; that is, the principle assumes no interference effect between the bets in the package, which means a rational agent's valuation for a package of bets on mutually exclusive propositions is a linear combination of her values for bets on the individual propositions. The standard model of rational choice in decision-making is the process of determining what options are available and then choosing the most preferred one according to some consistent criterion. On the individual level, the agent's choice of action is only rational if and only if that action is associated with the highest expected utility based on the agent's degrees of belief (credence) and desire toward the outcomes (utility). Suppose there is a package of bets (bet 1, bet 2, bet 3), and their respective prices are x, y, z, then the sum of the price an agent is willing to pay for this package is (x+y+z). If the agents accept all the offers of buying these three bets for a price less than the sum, then the agent's choice of action is considered to be irrational if she accepted such a bad deal instead of maximizing her interest, or she has other ulterior motives that are not characterized in this setting (that she can accomplish more by accepting this disadvantaged deal). In another case, if the agent 2 chooses to pay more than (x+y+z) for a set of bets valued less, her choice action is not rational either, as she did not maximize her interests. In these two cases, the agent’s preferences are only incoherent, or the values the agent set on the offered bets are not independent. Therefore, in the standard rational choice model, we assume the agent is rational (an expected utility maximizer) and would follow the package principle. Lara Buchak raises some questions about the soundness of the Sure-thing principle when explaining the Allais paradox. She proposes a risk-weighted expected utility model instead as the alternative to the expected utility model as she believes it would provide better justification for people's decision behavior. The Allais Paradox presents two choice situations: the choice between prospect A and B and the choice between prospect A' and B' .'According to the Surething principle, if two action yield the same outcomes conditional on event E, then the agent's preference between the two actions matches the agent's conditional preference between 2 actions given ~E (Savage). Only in this way the agent's preference over the choices of actions is consistent and context-independent. This axiom implies the critical property of expected utility: the separability of preferences across the mutually exclusive event. Accordingly, the agent should be indifferent between these two choice situations (AB, A'B') and have identical preferences in these two choice situations. However, the uncertainty of becoming a millionaire encourages people to prefer choice situation 1 (with action AB) rather than be indifferent between the two packages supported by the Sure-thing principle; more specifically, identify with "Allais preference" choose B' OVER A' and choose A over B) as choosing action A rather than B will make she become a millionaire with the probability of 1. Table 1 Allais Paradox (1953) Actions P = 0.01 States of the World P = 0.10 P = 0.89 3 A 1 million 1 million 1 million B 0 million 5 million 1 million A’ 1 million 1 million 0 B’ 0 million 5 million 0 The tendency of preferring a safer choice or a choice with less uncertainty, as described above, would sometimes contradict the conclusion reached by expected utility maximization as it added one extra factor (risk attitude) that is not included in expected utility theory (only considers the utility of outcomes and subjective belief about the probability of those outcomes conditional on the action being chosen). As a decision-making guideline, the Sure-thing principle suggests that the agent doesn't need to consider uncertainty in a decision that would not impact the decision anyway. In the Allais paradox, the agent's preference is influenced by the uncertainty of not being a millionaire, which makes the first-choice situation more appealing despite the Sure-thing principle sustained that the two-choice situations should be identical in the eyes of rational agents. Action B is less appealing when accompanied by action A with absolute certainty; therefore, as Risk-weighted expected utility model challenged the Sure-thing principle, it challenged the assumption that the value of a conjunction of two bets is not merely a sum of the value of the conjuncts. I would argue that she is right to reject the package principle because the interference effect between the bet's prizes significantly impacts our decision in numerous real-life situations. For all the circumstances when the agents do not value money linearly, the package principle is not applicable. For instance, if considering three bets, each valued at $500, and the agent's utility depends on whether she can buy a new iPhone through gambling (if the money earned does not reach $1200 in the end, then the agent's utility is zero). These three bets mean nothing to the agent individually, and only their combination can affect the agent's welfare; the value of the two bets combined is not the sum of the two bets' values. Another example would be the famous St. 4 Petersburg paradox. In the St. Petersburg game, the payoff is based on infinite coin tosses and infinite expected winnings. Consider each coin toss as a bet, then the sum of each bet would be infinitely large. However, no rational human will risk a large finite amount to play the game if she knows that the probability of getting an infinite amount of money is extremely small. This common recognition reflected human risk attitude could be used as a criterion for qualifying rational action. In another case, if the decision-making situation is sequential, then what the agent gained in the first round of the bet can affect its valuation in the second bet. Thus the total gain of the utility of a sequence of bets is not the sum of all utility gain from all bets (Joyce). The center of contention between the EU model and the risk-weighted expected utility model is the criteria of rationality. Should the individual risk assessment be included in the decision-making process, or should this extra piece of evidence be judged as the indicator of irrationality or rationality? One can argue that the if we do not value individual bets as their true value (that generated by the expected utility model), then the incoherent values will make the agents' preference inconsistent and therefore increase her Dutch bookability and lead the agent to less benefit. Depend on whether we want to use decision theory to identify a better choice of action or want to demystify particular decision pattern. If rationality is defined by whether the agent made the best decision dependent on all the evidence she had at the time and the purpose of the model is to approximate real-life human decision, then the rejection of the package principle added another layer of complexity to the model and improved our understanding about human decision patterns. As a solution to inconsistent preference led by not following the package principle, the agents were not told at the start that the decision problem was one in which a sequence of 5 opportunity sets was to be presented over time, then we would not be forced to judge the undesirable series of actions as irrational, thus avoid been money pumped. Works Cited Hájek, Alan. “CHAPTER 7 Dutch Book Arguments.” The Handbook of Rational and Social Choice, edited by Paul Anand et al., Oxford University Press, 2009, p. 0. Silverchair, https://doi.org/10.1093/acprof:oso/9780199290420.003.0008. Joyce, James. The Sure-Thing Principle and the Concept of Conditional Preference. Savage, Leonard J. The Foundations of Statistics. 2nd Revised ed. edition, Dover Publications, 1972.
