One stake one vote. Marc Fleurbaey. December Abstract. The weighted majority rule, which is widely applied much more than the


 Augustine Knight
 2 years ago
 Views:
Transcription
1 One stake one vote Marc Fleurbaey December 2008 Abstract The weighted majority rule, which is widely applied much more than the simple majority rule has good properties when the weights reflect the stakes of the voters in the decision. These properties can be contrasted with classical problems of democratic theory which are associated with the one person, one vote principle and the simple majority rule: tyranny of the majority, intensity problem, and many voting paradoxes. These observations suggest that the egalitarian one person, one vote principle should be replaced in democratic theory by a more fundamental proportionality principle. A new measure of sensitivity of a voting rule to the population s preferences is proposed, which is useful in determining the optimal voting weights of representatives. Keywords: majority rule, weighted majority, voting game, Banzhaf measure, power, democratic theory. This paper has benefited from a seminar at the LSE, from advice and comments by G. Arrhenius, N. Baigent, C. Beisbart, L. Bovens, R. Bradley, S. Brams, R. Goodin, A. Laruelle, C. List, K. Spiekermann, A. Voorhoeve, and from discussions with H. Brighouse, J. Roemer. CNRSCERSES, University ParisDescartes, LSE and IDEP. 1
2 it would be helpful for the development of democratic theory if we could assume that some means exist for comparing intensities of preferences (R. Dahl, 1956, p. 99) 1 Introduction The egalitarian principle one person, one vote is a cornerstone of democratic theory. It is intimately linked to the simple majority rule, which, in one form or another, plays an important role in many democratic institutions. In modern political thought, the core of the notion of democracy is its etymological core rule by the people which translates most naturally as majority rule if there are divisions of opinion. (Hardin 1993, p. 158). Butdemocratictheoryisplaguedwithdifficulties, many of which stem from undesirable properties of the simple majority rule: in particular, majorities can by cyclic (intransitivity), tyrannical (oppression of minorities), and inefficient (a combination of separate majoritarian decisions can be bad for all). In this paper it is observed that the weighted majority rule, in which each voter has a certain number of votes, has some nice properties if the weights are well chosen, i.e., if they are proportional to the voters stakes in the decisions. These properties suggest that the classical difficulties of democratic theory can be alleviated in this way, and therefore this paper argues for substituting a proportionality principle, one stake, one vote, to the egalitarian one person, one vote principle. Moreover, as argued in this paper, the perspectives of applications of weighted procedures are much greater than usually thought, in particular because they are already widely applied in various ways. The possibility to designing the voting rule on the basis of interpersonally comparable 2
3 information is traditionally excluded from the literature on voting systems, which, as will be argued here, appears unduly restrictive compared to actual practice. In order to assess the actual or potential merits of weighted voting rules in practice, it is necessary to analyze the choice of weights in some detail. In this paper, Sections 26 study how to determine the optimal weights for the weighted majority rule, in various informational contexts and at various levels of decision, when the objective is to maximize the (expected) sum of individual indices measuring individual interests. Some of the results extend existing results about the simple majority rule, while others are more original. Specifically, Section 2 presents a basic result that links weighted majority to the maximization of the sum of individual indices. Section 3 makes use of this result to solve the constitutional question of choosing a voting rule when one knows the relative stakes of different individuals but not how they vote. This provides a direct extension, and new proof, of the classical RaeTaylor theorem that is limited to the case of equal intensities. Section 4 extends the analysis to the case when the stakes are not known with certainty and when several issues with different stakes will have to be treated with the same voting rule. Section 5 briefly explores the intriguing case of voters who do not vote in their best interest, preparing the stage for Section 6, which is about the allocation of power across jurisdictions whose representatives may be more or less faithful to their constituencies. Taking stock, Section 7 explains how such results highlight a particular interpretation of the underlying causes of the classical difficulties of democratic theory. Section 8 discusses the practical relevance and the possible applications of the proportionality principle. Section 9 concludes. This paper relates to various branches of the literature. First, there are many studies of the social optimality of various classical voting rules (e.g., Bordley 1983, Merrill 1984, Beisbart et al. 2005, Barbera and Jackson 2006, Beisbart and Bovens 2007, Bovens and 3
4 Hartmann 2008). Although they all rely on specific assumptions about utilities, these studies tend to downplay the role of these informational assumptions and do not argue directly in favor of eliciting utility information in order to improve the voting rule. In these studies the weighted majority rule is considered only for representatives of unequally sized constituencies. Bordley (1985, 1986) came closer to the idea defended in this paper by arguing in favor of weighted majority rule for ordinary voters, with weights derived from specific utility assumptions. However, he also downplayed the role of utility assumptions, instead emphasizing the role of the observed distribution of votes in the definition of weights, focusing on correlations between individual utilities within factions. This paper proposes a more direct study of the idea of making use of utility information, highlighting cases in which weighted majority is the optimal rule. Another related branch of the literature is interested in letting individuals allocate a budget of voting weights or points between several issues and studies the advantages of such games over more rigid voting rules (Casella 2005, Casella et al. 2006, HortalaVallve 2007, Jackson and Sonnenschein 2007). Such complex voting games are not studied here but will be briefly discussed in Section 4. 2 Welfare maximum In this section we provide a simple extension of the wellknown fact that the simple majority rule maximizes the sum of utilities under the assumption of equal intensity across voters. This basic result is a key lemma for our analysis. The framework is as simple as possible and is borrowed from the theory of voting games and of power measurement. 1 There is a fixed population N = {1,..., n}. A simple 1 Excellent syntheses can be found in Mueller (2003) for the general literature on voting and in Felsenthal and Machover (1998) for voting games and power measurement. See also Laruelle and Valenciano 4
5 voting game is a set V of subsets of N called winning coalitions. 2 Concretely, one has to imagine that when a binary decision (yes or no) is on the table, one collects the votes, and if the set of yes voters is a winning coalition, the yes decision wins. 3 To shorten expressions in subscripts, let A i denote A\{i} and A+i denote A {i}. A coalition that is not winning (A / V) is called a losing coalition. The size of a coalition A is denoted #A. ThesimplemajorityruleisassociatedtothesimplevotinggameM such that A M if and only if #A >n/2. A weighted majority rule, with weights (w 1,..., w n ) R n, is associated to the simple voting game W such that A W if and only if P i A w i > P i N w i/2, i.e., the members of A have a majority of the total weights. 4 Obviously, the simple majority rule is a particular case of weighted majority rule, in which w i =1for all i N (one person, one vote). 5 Consider a given binary decision, yes or no. Each voter i N has a utility U + i with the favorable decision for him, and a utility U i with the unfavorable decision for him. Let i = U + i U i. The utility derived by voter i from a particular alternative, say yes, (2008). The paper is selfcontained in the sense that it does not assume prior knowledge of this theory from the reader. 2 In the literature, a simple voting game is required to satisfy restrictions: N V, / V, and if A V and A B, then B V. The class considered in this paper is larger. 3 This description, under the definition of a simple voting game, does not exclude the possibility that the set of no voters is also a winning coalition the procedure as described is biased in favor of the yes decision. A proper voting game would not have disjoint winning coalitions. Unless otherwise indicated, we do not restrict attention to proper voting games, but nothing of substance would be changed if we did. The weighted majority rule examined in this paper is a proper voting game. 4 The weights can be negative, in which case the weighted majority game is not a simple voting game in the usual sense because it may happen that A W, A B and B/ W. 5 A more satisfactory description of these rules would yield a neutral decision (i.e., tossing a coin) instead of a no decision when a vote is obtained. Again, apart from expositional complications nothing of substance would be changed. 5
6 is denoted U i (yes). Suppose we are interested in maximizing social welfare, understood as the sum of individual utilities, P i N U i. This sum need not represent the classical utilitarian criterion because the numbers U i can be interpreted in many ways. The most convenient for our purposes is to understand them as measuring the social value of i s situation for the observer. This makes it possible for the welfare criterion to incorporate a priority for the worstoff as well as nonwelfarist considerations in the measurement of individual wellbeing. Any continuous separable social criterion, welfarist or nonwelfarist, is encompassed in our approach. The term utility is retained only for its simplicity and in order to make the results more easily comparable with those found in the literature. As announced, the following result extends the observation that the simple majority rule maximizes the sum of utilities in the case of equal intensity: the weighted majority rule maximizes the sum of utilities when voters have unequal intensities and their weights are proportional to the utility differences i. It is assumed that each voter votes for the option which is more favorable to him (this assumption will be relaxed in Section 5). The indifferent voters ( i =0) vote in an undetermined manner (they vote either yes or no, or can even abstain), this does not affecttheoutcomesincetheirvoteisnotcountedby the weighted majority rule in the case considered in the proposition. Proposition 1 If the weights w i are proportional to i, the weighted majority rule selects the decision which yields the greater sum of utilities. Proof. Suppose that yes yields a greater sum of utilities: P i N U i(yes) > P i N U i(no). Putting the yes voters on the left side of the inequality and the no voters on the right, one obtains (the yes and no voters may include some indifferent voters, but they can be 6
7 ignored in the formula because for them U i (yes) U i (no) =0): This also reads i N: U i (yes)>u i (no) U i (yes) U i (no) > i N: U i (yes)>u i (no) and by assumption is equivalent to i N: U i (yes)>u i (no) i > w i > i N: U i (yes)<u i (no) i N: U i (yes)<u i (no) i N: U i (yes)<u i (no) which implies that the weighted majority rule will select yes. U i (no) U i (yes). A symmetric argument applies if P i N U i(yes) < P i N U i(no). i w i, 3 Constitutionmaking According to the RaeTaylor theorem (Rae 1969, Taylor 1969), the simple majority rule minimizes the risk that a voter disagrees with the collective decision, over the family of qualified majority rules. 6 As a result, it maximizes the expected utility of a constitutionmaker who does not know in advance on which side of the issue he will be falling, or which voter he will be, on the assumption that all voters have an equal intensity of preferences over the issue. This section is about the fact that if the equal intensity assumption is relaxed, the weighted majority rule is singled out instead of the simple majority rule. A much simpler proof of the RaeTaylor theorem is a byproduct of this analysis. A word on the interpretation of uncertainty is in order here. Most of the literature seems to consider that the constitutionmaker is a real individual who either does not yet 6 Aqualified majority involves a threshold that differs from n/2. This theorem has been generalized by Straffin (1977) to all simple voting games. See also Curtis (1972), Dubey and Shapley (1979). 7
8 know what his interests will be, or decides behind a veil of ignorance which artificially renders him impartial. The former interpretation is not satisfactory because the situation of full uncertainty is never obtained in practice, as one always knows some key aspects of one s present and future preferences. The latter has been contested, because even if choices made behind a veil of ignorance are certainly impartial, they need not be fair in a more substantial sense. Behind a veil of ignorance, one may be willing to take a risk for oneself in order to obtain greater advantages for oneself in other states of nature translated into distributive judgments, this means that minorities can be sacrificed for the sake of greater pleasures of the rest of the population: it is dubious that the tradeoffs in the former context are relevant guidelines for tradeoffs in the latter. A different interpretation of uncertainty is proposed here. What we are after in constitutionmaking is a set of institutions that produce good outcomes on average over the possible cases that may occur. The formalism of uncertainty is adopted for convenience, but really we simply want to have an idea of the average performance of the considered rules. The results below will use the phrase on average while using the formalism of expected value. Consider a single binary decision again. Suppose first that uncertainty is limited to the decision (yes or no) that each voter will favor, whereas U + i,u i and therefore i is already known for every i N. Uncertainty about i is studied in the next section. Assuming that we know the voters intensities without knowing the direction of their preferences corresponds to a special but relevant case, as we often have indications about intensities by objective proxies (location, occupation, financial situation) whereas the direction of preferences may depend on beliefs or political views which are not observable. There is a given probability law over the 2 n possible coalitions of yes voters, with p A denoting the probability of coalition A N. 7 Once a simple voting game V has been 7 This general probabilistic description, which does not assume that voters are independent, is also found in Laruelle and Valenciano (2005). A similar kind of generalization can also be found in Curtis 8
9 picked, one can compute the expected utility EU i of each voter. Let P(e) denote the probability of event e. One has: EU i = U + i P(i agrees with the decision)+u i P(i disagrees with the decision) = i P(i agrees with the decision)+u i. (1) In particular, P(i agrees with the decision) = P(i votes yes and yes wins) +P (i votes no and no wins) = A V: i A p A + A/ V: i/ A p A. (2) We seek a rule that maximizes the sum of utilities on average, i.e., that maximizes E P i N U i. Theansweristhesameasintheprevioussection,forasimplereason:whatever the profile of preferences of the voters, the weighted majority rule selects the alternative which yields the greater sum of utilities. In addition, it is shown that this is essentially the only rule that maximizes the sum of utilities on average: other acceptable rules may differ only on the decision made for ties. Proposition 2 If the weights w i are proportional to i, theweightedmajorityrulemaximizes the sum of utilities on average, over the whole set of simple voting games. Conversely, if p A > 0 for all A N, any simple voting game maximizing the sum of utilities on average is such that every coalition A for which P i A i > P i N i/2 is a winning coalition, and no coalition A for which P i A i < P i N i/2 is a winning coalition. (1972), who restricts attention to symmetric games, seeks to minimize the expected number of disappointed voters, and assumes a general probability distribution over the possible numbers (not the coalitions) of yes voters. 9
10 Proof. By Prop. 1, we know that for each coalition of yes voters, the weighted majority rule selects the option with greater P i N U i. Thereforenosimplevotinggamecanyield agreatere P i N U i. For the converse, let V be any simple voting game. If P i A i > P i N i/2 and A is the coalition of yes voters, then yes is the decision yielding greater total utility. Since p A > 0, one should retain A as a winning coalition for V. Symmetrically, if P i A i < P i N i/2 and A is the coalition of yes voters, then no is the decision yielding greater total utility. One should not retain A as a winning coalition for V. Failure to satisfy any of these two properties reduces total expected utility if p A > 0 for all A N. Note that the argument in the proof applies for any probability law defined over the coalitions of yes voters, which is much more general than the standard formulations of the RaeTaylor theorem in which either equally probable coalitions or independence between voters is assumed. And this argument provides a much simpler way to prove the Rae Taylor theorem than the standard proofs which involve a computation of the probability P(i agrees with the decision) (as in Straffin 1977) or P(i disagrees with the decision) (as in Taylor 1969). In fact, it is also easy to prove the result directly by looking at the probabilities but with very little algebra. Maximizing the expected sum of utilities is equivalent to maximizing the sum of expected utilities, which itself is equivalent, by (1), to maximizing P i N ip(i agrees with the decision). One computes: i P(i agrees with the decision) = i p A + p A i N i N A V: A/ V: i A i/ A = Ã! p A i + Ã! p A i. A V: i A A/ V: i/ A One sees that moving a losing coalition A into V changes the total sum by Ã p A i! i, i A i/ A 10
11 which is beneficial if and only if P i A i > P i/ A i. One should then put into V exactly the coalitions satisfying this condition. This is precisely what W does. Theargumentinthefirst proof above is in fact more powerful than this second one because the former also implies that the weighted majority rule maximizes Ef P i N U i for any arbitrary increasing function f. This means that the weighted majority rule is not only good in terms of average performance, but in every aspect of the distribution of performance. Let us now restrict attention to the uniform law (p A =1/2 n for all A), i.e., the case in which every voter has an equal chance of voting yes or no and votes independently of the others. This particular case is central in the literature on voting power and will enable us to relate the above result to some wellknown results about the Banzhaf measure of voting power. The Banzhaf measure can be defined as the probability that i is critical, i.e. that yes wins if i votes yes and no wins if i votes no. The probability is usually computed under the assumption that all coalitions of yes voters are equally likely. This makes sense if the goal is to measure the structural power of a voter in the game form rather than his effective influence given the likely preference profile of the population. Therefore the Banzhaf measure is simply the number η i of winning coalitions in which i is critical divided by the total number of possible coalitions to which i may belong, 2 n 1. Following standard conventions, let β 0 i denote the Banzhaf measure of power for voter i in a given simple voting game: β 0 i = η i /2 n 1. The sum of these numbers over the population, P i N β0 i, is commonly used as a measure of the sensitivity of the simple voting game to the preferences of the population. A result closely related to the RaeTaylor theorem is that the simple majority rule maximizes sensitivity over the whole set of simple voting games. The relation between the two results comes from the following fact, which states a simple link between the probability for a voter of being critical with the probability of 11
12 his agreeing with the decision. 8 Lemma 1 Under the uniform probability assumption, P(i is critical) =2P(i agrees with the decision) 1. Proof. Let ω denote the number of winning coalitions and ω i the number of winning coalitions containing i. The number of nonwinning coalitions not containing i equals 2 n 1 (ω ω i ) i.e., the number of coalitions not containing i minus the number of winning coalitions not containing i. From equation (2) one computes P(i agrees with the decision) = 1 2 n ωi +2 n 1 (ω ω i ). The number of coalitions in which i is critical, η i, equals the number of winning coalitions containing i minus the number of winning coalitions in which i is not critical. The latter number equals the number of winning coalitions not containing i (adding i to any such coalition gives a winning coalition in which i is not critical). Therefore one has: P(i is critical) = 1 2 n 1 [ω i (ω ω i )]. Simple algebraic manipulation yields the desired relation between the two probabilities. The result about majority rule and sensitivity P i N β0 i can therefore be extended: the weighted majority rule maximizes P i N iβ 0 i and is almost characterized by this property. This result is not particularly interesting in itself because P i N iβ 0 i is not an immediately intuitive objective, but it will be extended and explained in the comment that follows. 9 8 It is attributed to Penrose (1946) by Felsenthal and Machover (1998). See also Brams and Lake (1978), Straffin (1978), Dubey and Shapley (1979) and Laruelle et al. (2006) for further developments. 9 This result is also mentioned and proved in Beisbart and Bovens (2007, Theorem 1). 12
13 Proposition 3 Under the uniform probability assumption, if the weights w i are proportional to i, the weighted majority rule maximizes P i N iβ 0 i overthewholesetofsimple voting games. Conversely, any simple voting game maximizing P i N iβ 0 i is such that every coalition A for which P i A i > P i N i/2 is a winning coalition, and no coalition A for which P i A i < P i N i/2 is a winning coalition. Proof. This is an immediate corollary of Prop. 2, in view of Lemma 1. Here is another, direct, proof. Let 1[e] denote the function which equals 1 if e is true and 0 otherwise. In the proof of Lemma 1 it is shown that η i =2ω i ω. Onetherefore has i η i = 2 i ω i i ω i N i N i N = 2 i 1[i A] i 1 i N A V: i N A V = Ã 2 i! i. A V i A i N In order to maximize this sum, one would ideally retain A if 2 P i A i P i N i > 0 and exclude A if 2 P i A i P i N i < 0, as done by W. The validity of Proposition 3 cannot be extended much beyond the case of uniform probability. It is possible to generalize the Banzhaf measure to any given probability law over the formation of coalitions 10 but the above result does not typically hold for this generalized measure of power. It is, however, possible to extend Proposition 3 to the 10 As proposed in Laruelle and Valenciano (2005). 13
14 general case of probability p if sensitivity is redefined as σ = i N i [P(i agrees) P(i disagrees)] = i i N = A V: A V: i A p A Ã i A p A + A/ V: i/ A i i/ A p A A V: i/ A! i A/ V: p A A/ V: i A p A Ã i A p A i i/ A i!. Indeed, EU i = U + i P(i agrees with the decision)+u i P(i disagrees with the decision) = i P(i agrees with the decision)+u i = U + i i P(i disagrees with the decision), so that 2EU i = U + i + U i + i [P(i agrees) P(i disagrees)]. From this equation it is clear that Proposition 3 holds in the general case when sensitivity is redefined as suggested. In the uniform probability case, from Lemma 1 one has P(i agrees) P(i disagrees) =P(i is critical), implying that the two definitions of sensitivity are then identical. 11 But in the general case it appears that the new notion σ is superior. Indeed, what is important, if one wants the decision to be sensitive to the voters preferences, is not so much that they should have a maximal probability of being critical, but that they should have a maximal expected 11 Laruelle et al. (2006), who argue that agreeing with the decision is more important than being critical, show that this equality holds for all voters and all voting games only inthecaseofuniform probability. Curtis (1972) and Brams and Lake (1978) study the average value of P(i agrees) over all voters, which is linearly related to the measure of sensitivity proposed here. 14
15 utility, which depends on the probability of agreeing, or equivalently, on the gap between the probability of agreeing and the probability of disagreeing. Whether they agree by being critical or not is not so relevant. This notion of sensitivity will turn out to be relevant in the analysis of representative democracy in Section 7. 4 Multiple issues and uncertainty about stakes In the previous section, we have considered a single binary decision and assumed that the vector of stakes ( 1,..., n ) is known. If one extends the veil of ignorance and assumes that many binary decisions k = 1,...,m will be taken in a sequence, with additively separable utilities over the various decisions (i.e., U i = U 1 i +...+U m i ) and a priori unknown ( k 1,..., k n) for each specific decision k, the results are untouched if one assumes that the simple voting game can be adapted to each decision k and that ( k 1,..., k n) is disclosed in due time before each decision. In this case taking the weighted majority rule with weights proportional to ( k 1,..., k n) for each decision k is optimal. More interesting is the case when the same simple voting game has to be kept throughout the whole sequence of future decisions. One must be precise about the probabilistic setting. To begin with, let us assume that for each issue k, the value of the pair (Ui k,ui k+ ) is drawn from a probability law that may be specific toi and k, independently of the values for other issues. The same probability law p defines the probability of coalitions of yes voters over every issue (the probability of i voting yes or no over some issue is assumed to be independent of the values of Ui k,ui k+ and of his vote on other issues). Let E U denote expectation with respect to possible values of utilities, and E A 15
16 denote expectation with respect to possible coalitions over all issues. One computes m E U E A U i = E U E A Ui k = k=1 m k=1 E U k i P(i agrees with decision k)+ui k. Because the same simple voting game is operating for all k and the same probability law governs the formation of coalitions over all issues, P(i agrees with decision k) is a constant P(i agrees with decision), so the formula simplifies into m m E U E A U i = EU k i P(i agrees with decision)+ E U Ui k. k=1 Similarity with the formulae relative to Proposition 2 implies that the weighted majority rule with weights proportional to P m P k=1 EU k i for i =1,..., n maximizes E i N U i.in other words, voter i s weight should be proportional to the expected value of the sum of hisstakesoverallissues. k=1 Proposition 4 If the weights w i are proportional to P m k=1 EU k i, the weighted majority rule maximizes the sum of utilities on average, over the whole set of simple voting games. Conversely, if p A > 0 for all A N, any simple voting game maximizing the sum of utilities on average is such that every coalition A for which P P m P P i A k=1 EU k i > m i N k=1 EU k i /2 is a winning coalition, and no coalition A for which P P m P P i A k=1 EU k i < m i N k=1 EU k i /2 is a winning coalition. This result covers the following two subcases: Subcase 1: The k i are known but the same simple voting game must be retained for all k in this subcase the average value of k i over all values of k defines a suitable weigthing scheme. Subcase 2: There is only one issue but uncertainty prevails about i in this subcase theexpectedvalueof i defines a suitable weighting scheme. 16
17 A situation that is likely to occur in many applications is that a clear separation will be observed between two subpopulations. In one subpopulation, the individuals will have high and roughly equal values of average expected stake, whereas this value is very low for the members of the other subpopulation. Such a situation may be observed in particular for the issues affecting a local population (e.g., inhabitants of a town or any particular geographical district), the outsiders being largely unaffected on average by such issues. In this case a reasonable application of the above result is to restrict voting rights on such local issues to the local population, and grant them equal weights. A generalization to the case when each issue k has a specific probability distribution p k over coalitions is possible but one no longer obtains an ordinary weighted majority rule. One then computes m E U k i P(i agrees with decision k) = k=1 m E U k i k=1 = A V m k=1 p k A A V: i A p k A + A/ V: i/ A i A E U k i + A/ V p k A m p k A k=1 i/ A E U k i Moving a losing coalition A into V changes this expression by the amount Ã m p k A E U k i! E U k i = m p k AE U k i m p k AE U k i. k=1 i A i/ A i A k=1 i/ A k=1 We should therefore make A a winning coalition if this term is positive, and a losing coalition if it is negative. This yields a voting game in which the weight of a voter does not depend on the issue itself but depends on the coalition of yes voters 12 for the current issue: when this coalition is A, voter i s weight is P m k=1 pk A E U k i. This means that voter i s weight is greater, other things equal, when A is more likely to be the coalition of yes voters over issues for which his expected stakes are greater Voting rules with weights depending on the realized votes are mentioned in particular in Bordley (1986) and Barbera and Jackson (2006). 13 One may raise the second best question of what the optimal weights would be for weighted majority 17
18 When the distribution of utilities is not independent on the distribution of coalitions of yes voters, one obtains a similar result in which voter i s weight is P m k=1 pk AE U/A k i, where E U/A denotes the expected value conditional on A being the coalition of yes voters. Proposition 5 Inthefullygeneralcaseinwhicheachissuehasaspecific probability distribution p k and uncertain utilities for an issue may depend on the realized coalition, the following voting game maximizes the sum of utilities on average, over the whole set of simple voting games: a coalition is winning if and only if wi A > wi A /2, i A i N where wi A = P m k=1 pk AE U/A k i. Conversely, any simple voting game maximizing the sum of utilities on average is such that every coalition A for which P i A wa i > P i N wa i /2 is a winning coalition, and no coalition A for which P i A wa i < P i N wa i /2 is a winning coalition. Contrary to Proposition 2, Propositions 4 and 5 cannot be generalized to the maximization of Ef P i N U P i because there is no voting game that maximizes i N U i for every configuration of utilities. The situation is even bleak in some configurations of utilities because the constraint to have the same game over all issues can generate bad decisions. Consider a threeagent, twoissue situation in which the k i areasinthefollowing table (a positive number means the agent votes yes, and vice versa). 1 i 2 i Ann +31 Bob Chris 11 rule in this context. Simulations done by the author suggest that, although weights proportional to P m k=1 E U k i are no longer optimal in general, they are typically close to optimal weights. This issue is left for future research. 18
19 In this configuration of utilities, the optimal decision (maximizing the sum of utilities) is yes for both issues. But if the probability distribution of coalitions is the same for both issues, it is optimal to use a weighted majority rule, and by symmetry Ann and Bob must get the same weight. The consequence is that they neutralize each other over every issue and no wins in both decisions. It is possible to obtain a yes decision over one issue by giving unequal weights to Ann and Bob (in violation of impartiality), but it is impossible to obtain a yes decision over the two issues. This kind of example highlights the importance of defining specific weightsforeach issue separately. Interestingly, in some cases the specific weightscan bemadetorelyon voters private information, as shown in the literature on more complex voting games in which voters can allocate points between issues (e.g., Casella 2005, Jackson and Sonnenschein2007). IfAnnandBobwereassignedfourpointsandChristwopoints,Annand Bob could secure a yes vote on their most important issue by putting at least three of their pointsonit.toachievethisoutcomethedesigneroftherulemustknowthatann sand Bob sstakesareonaveragegreaterthanchris sbutneednotknowwhichissueismore important for either of them. More generally, granting a number of points proportional P to E m U k=1 k i to each i and letting i allocate this quantity across issues may be an interesting way of improving upon the simple voting games considered in this paper. 5 Erratic voters Now let us introduce the complication that some voters may vote against their interests as measured by U i. This may be due to a discrepancy between U i and their preferences (e.g., if U i is a nonwelfarist measure), or to an informational problem that gives them mistaken beliefs about their interests. We first consider the simple setting of Section 2 and assume that N is partitioned 19
20 into N p (the prointerest voters who always vote in their interest) and N c (the contrainterest voters who always vote against their interest). Now, maximizing the sum of utilities requires giving the latter a negative weight. This result has no practical interest, if only because negative weights make the voting rule highly manipulable. But it is useful as a theoretical clarification and as a preparation for the next section. Proposition 6 If the weights w i are proportional to i for i N p and to i for i N c, the weighted majority rule selects the decision which yields the greater sum of utilities. Proof. Suppose P i N U i(yes) > P i N U i(no). Putting the yes voters on the left side of theinequalityandthenovotersontheright,asintheproofofprop.1,nowyields i N p : U i (yes)>u i (no) i i N c : U i (yes)<u i (no) i > i N p : U i (yes)<u i (no) i i N c : U i (yes)>u i (no) i. If we turn to the framework of Section 3, we may assume that there is a probability law such that p A,B is the probability that A is the coalition of voters whose interest is yes and B is the coalition of yes voters. An agent votes in his interest ( pro ) if i A B or i N \ (A B). We will say that voter i is steady when the following conditional probability is the same for all B N : P(i is pro B vote yes) = A N p A,B i A A N i/ A A N p A,B A N p A,B p A,B if i B if i/ B. The next result defines optimal weights which are jointly proportional to the stakes and to an index of reliability of the voter. This index, equal to 2P(i pro) 1, goes from 1 for a voter who consistently votes in his interest to 1 foravoterwhoalwaysvotes against his interest. 14 It is zero for a voter who has an equal chance of voting pro or 14 There is some similarity between this result and the Condorcet jury theorem where badly informed members of the jury should have a negative weight (BenYashar and Nitzan 1997). 20
LOOKING FOR A GOOD TIME TO BET
LOOKING FOR A GOOD TIME TO BET LAURENT SERLET Abstract. Suppose that the cards of a well shuffled deck of cards are turned up one after another. At any timebut once only you may bet that the next card
More informationarxiv:1112.0829v1 [math.pr] 5 Dec 2011
How Not to Win a Million Dollars: A Counterexample to a Conjecture of L. Breiman Thomas P. Hayes arxiv:1112.0829v1 [math.pr] 5 Dec 2011 Abstract Consider a gambling game in which we are allowed to repeatedly
More informationMoral Hazard. Itay Goldstein. Wharton School, University of Pennsylvania
Moral Hazard Itay Goldstein Wharton School, University of Pennsylvania 1 PrincipalAgent Problem Basic problem in corporate finance: separation of ownership and control: o The owners of the firm are typically
More informationChapter 7. Sealedbid Auctions
Chapter 7 Sealedbid Auctions An auction is a procedure used for selling and buying items by offering them up for bid. Auctions are often used to sell objects that have a variable price (for example oil)
More informationThe Math. P (x) = 5! = 1 2 3 4 5 = 120.
The Math Suppose there are n experiments, and the probability that someone gets the right answer on any given experiment is p. So in the first example above, n = 5 and p = 0.2. Let X be the number of correct
More informationP (A) = lim P (A) = N(A)/N,
1.1 Probability, Relative Frequency and Classical Definition. Probability is the study of random or nondeterministic experiments. Suppose an experiment can be repeated any number of times, so that we
More information1 Uncertainty and Preferences
In this chapter, we present the theory of consumer preferences on risky outcomes. The theory is then applied to study the demand for insurance. Consider the following story. John wants to mail a package
More informationA New Interpretation of Information Rate
A New Interpretation of Information Rate reproduced with permission of AT&T By J. L. Kelly, jr. (Manuscript received March 2, 956) If the input symbols to a communication channel represent the outcomes
More informationPascal is here expressing a kind of skepticism about the ability of human reason to deliver an answer to this question.
Pascal s wager So far we have discussed a number of arguments for or against the existence of God. In the reading for today, Pascal asks not Does God exist? but Should we believe in God? What is distinctive
More informationMAT2400 Analysis I. A brief introduction to proofs, sets, and functions
MAT2400 Analysis I A brief introduction to proofs, sets, and functions In Analysis I there is a lot of manipulations with sets and functions. It is probably also the first course where you have to take
More informationECON 459 Game Theory. Lecture Notes Auctions. Luca Anderlini Spring 2015
ECON 459 Game Theory Lecture Notes Auctions Luca Anderlini Spring 2015 These notes have been used before. If you can still spot any errors or have any suggestions for improvement, please let me know. 1
More informationTwoState Options. John Norstad. jnorstad@northwestern.edu http://www.norstad.org. January 12, 1999 Updated: November 3, 2011.
TwoState Options John Norstad jnorstad@northwestern.edu http://www.norstad.org January 12, 1999 Updated: November 3, 2011 Abstract How options are priced when the underlying asset has only two possible
More informationWhy is Insurance Good? An Example Jon Bakija, Williams College (Revised October 2013)
Why is Insurance Good? An Example Jon Bakija, Williams College (Revised October 2013) Introduction The United States government is, to a rough approximation, an insurance company with an army. 1 That is
More informationTest of Hypotheses. Since the NeymanPearson approach involves two statistical hypotheses, one has to decide which one
Test of Hypotheses Hypothesis, Test Statistic, and Rejection Region Imagine that you play a repeated Bernoulli game: you win $1 if head and lose $1 if tail. After 10 plays, you lost $2 in net (4 heads
More informationPersuasion by Cheap Talk  Online Appendix
Persuasion by Cheap Talk  Online Appendix By ARCHISHMAN CHAKRABORTY AND RICK HARBAUGH Online appendix to Persuasion by Cheap Talk, American Economic Review Our results in the main text concern the case
More informationMINITAB ASSISTANT WHITE PAPER
MINITAB ASSISTANT WHITE PAPER This paper explains the research conducted by Minitab statisticians to develop the methods and data checks used in the Assistant in Minitab 17 Statistical Software. OneWay
More informationThe Relation between Two Present Value Formulae
James Ciecka, Gary Skoog, and Gerald Martin. 009. The Relation between Two Present Value Formulae. Journal of Legal Economics 15(): pp. 6174. The Relation between Two Present Value Formulae James E. Ciecka,
More information1. Introduction. 1 I thank Simon Hix for his invitation to write this response, and Robert Goodin for his helpful suggestions.
1 The Voting Power Approach: A Theory of Measurement A Response to Max Albert Christian List Australian National University and London School of Economics 1 Abstract. Max Albert (2003) has recently argued
More informationThe Basics of Graphical Models
The Basics of Graphical Models David M. Blei Columbia University October 3, 2015 Introduction These notes follow Chapter 2 of An Introduction to Probabilistic Graphical Models by Michael Jordan. Many figures
More information1. R In this and the next section we are going to study the properties of sequences of real numbers.
+a 1. R In this and the next section we are going to study the properties of sequences of real numbers. Definition 1.1. (Sequence) A sequence is a function with domain N. Example 1.2. A sequence of real
More informationQuiggin, J. (1991), Too many proposals pass the benefit cost test: comment, TOO MANY PROPOSALS PASS THE BENEFIT COST TEST  COMMENT.
Quiggin, J. (1991), Too many proposals pass the benefit cost test: comment, American Economic Review 81(5), 1446 9. TOO MANY PROPOSALS PASS THE BENEFIT COST TEST  COMMENT by John Quiggin University of
More information, for x = 0, 1, 2, 3,... (4.1) (1 + 1/n) n = 2.71828... b x /x! = e b, x=0
Chapter 4 The Poisson Distribution 4.1 The Fish Distribution? The Poisson distribution is named after SimeonDenis Poisson (1781 1840). In addition, poisson is French for fish. In this chapter we will
More informationIntroduction to. Hypothesis Testing CHAPTER LEARNING OBJECTIVES. 1 Identify the four steps of hypothesis testing.
Introduction to Hypothesis Testing CHAPTER 8 LEARNING OBJECTIVES After reading this chapter, you should be able to: 1 Identify the four steps of hypothesis testing. 2 Define null hypothesis, alternative
More informationInduction. Margaret M. Fleck. 10 October These notes cover mathematical induction and recursive definition
Induction Margaret M. Fleck 10 October 011 These notes cover mathematical induction and recursive definition 1 Introduction to induction At the start of the term, we saw the following formula for computing
More informationMATH10212 Linear Algebra. Systems of Linear Equations. Definition. An ndimensional vector is a row or a column of n numbers (or letters): a 1.
MATH10212 Linear Algebra Textbook: D. Poole, Linear Algebra: A Modern Introduction. Thompson, 2006. ISBN 0534405967. Systems of Linear Equations Definition. An ndimensional vector is a row or a column
More informationMath/Stats 425 Introduction to Probability. 1. Uncertainty and the axioms of probability
Math/Stats 425 Introduction to Probability 1. Uncertainty and the axioms of probability Processes in the real world are random if outcomes cannot be predicted with certainty. Example: coin tossing, stock
More informationWHAT ARE MATHEMATICAL PROOFS AND WHY THEY ARE IMPORTANT?
WHAT ARE MATHEMATICAL PROOFS AND WHY THEY ARE IMPORTANT? introduction Many students seem to have trouble with the notion of a mathematical proof. People that come to a course like Math 216, who certainly
More informationOPRE 6201 : 2. Simplex Method
OPRE 6201 : 2. Simplex Method 1 The Graphical Method: An Example Consider the following linear program: Max 4x 1 +3x 2 Subject to: 2x 1 +3x 2 6 (1) 3x 1 +2x 2 3 (2) 2x 2 5 (3) 2x 1 +x 2 4 (4) x 1, x 2
More informationProbability and Expected Value
Probability and Expected Value This handout provides an introduction to probability and expected value. Some of you may already be familiar with some of these topics. Probability and expected value are
More informationEconomics 1011a: Intermediate Microeconomics
Lecture 12: More Uncertainty Economics 1011a: Intermediate Microeconomics Lecture 12: More on Uncertainty Thursday, October 23, 2008 Last class we introduced choice under uncertainty. Today we will explore
More informationThe MarketClearing Model
Chapter 5 The MarketClearing Model Most of the models that we use in this book build on two common assumptions. First, we assume that there exist markets for all goods present in the economy, and that
More informationMathematics and Social Choice Theory. Topic 1 Voting systems and power indexes. 1.1 Weighted voting systems and yesno voting systems
Mathematics and Social Choice Theory Topic 1 Voting systems and power indexes 1.1 Weighted voting systems and yesno voting systems 1.2 Power indexes: ShapleyShubik index and Banzhaf index 1.3 Case studies
More informationTHE FUNDAMENTAL THEOREM OF ARBITRAGE PRICING
THE FUNDAMENTAL THEOREM OF ARBITRAGE PRICING 1. Introduction The BlackScholes theory, which is the main subject of this course and its sequel, is based on the Efficient Market Hypothesis, that arbitrages
More informationOn the Significance of the Absolute Margin Christian List British Journal for the Philosophy of Science, forthcoming
1 On the Significance of the Absolute Margin Christian List British Journal for the Philosophy of Science, forthcoming Abstract. Consider the hypothesis H that a defendant is guilty (a patient has condition
More information6 Scalar, Stochastic, Discrete Dynamic Systems
47 6 Scalar, Stochastic, Discrete Dynamic Systems Consider modeling a population of sandhill cranes in year n by the firstorder, deterministic recurrence equation y(n + 1) = Ry(n) where R = 1 + r = 1
More informationWeek 4: Gambler s ruin and bold play
Week 4: Gambler s ruin and bold play Random walk and Gambler s ruin. Imagine a walker moving along a line. At every unit of time, he makes a step left or right of exactly one of unit. So we can think that
More informationSome Polynomial Theorems. John Kennedy Mathematics Department Santa Monica College 1900 Pico Blvd. Santa Monica, CA 90405 rkennedy@ix.netcom.
Some Polynomial Theorems by John Kennedy Mathematics Department Santa Monica College 1900 Pico Blvd. Santa Monica, CA 90405 rkennedy@ix.netcom.com This paper contains a collection of 31 theorems, lemmas,
More informationChapter 15 Introduction to Linear Programming
Chapter 15 Introduction to Linear Programming An Introduction to Optimization Spring, 2014 WeiTa Chu 1 Brief History of Linear Programming The goal of linear programming is to determine the values of
More informationUnraveling versus Unraveling: A Memo on Competitive Equilibriums and Trade in Insurance Markets
Unraveling versus Unraveling: A Memo on Competitive Equilibriums and Trade in Insurance Markets Nathaniel Hendren January, 2014 Abstract Both Akerlof (1970) and Rothschild and Stiglitz (1976) show that
More informationOptimal Auctions Continued
Lecture 6 Optimal Auctions Continued 1 Recap Last week, we... Set up the Myerson auction environment: n riskneutral bidders independent types t i F i with support [, b i ] residual valuation of t 0 for
More informationRegret and Rejoicing Effects on Mixed Insurance *
Regret and Rejoicing Effects on Mixed Insurance * Yoichiro Fujii, Osaka Sangyo University Mahito Okura, Doshisha Women s College of Liberal Arts Yusuke Osaki, Osaka Sangyo University + Abstract This papers
More informationCryptography and Network Security Department of Computer Science and Engineering Indian Institute of Technology Kharagpur
Cryptography and Network Security Department of Computer Science and Engineering Indian Institute of Technology Kharagpur Module No. # 01 Lecture No. # 05 Classic Cryptosystems (Refer Slide Time: 00:42)
More informationMOP 2007 Black Group Integer Polynomials Yufei Zhao. Integer Polynomials. June 29, 2007 Yufei Zhao yufeiz@mit.edu
Integer Polynomials June 9, 007 Yufei Zhao yufeiz@mit.edu We will use Z[x] to denote the ring of polynomials with integer coefficients. We begin by summarizing some of the common approaches used in dealing
More informationALMOST COMMON PRIORS 1. INTRODUCTION
ALMOST COMMON PRIORS ZIV HELLMAN ABSTRACT. What happens when priors are not common? We introduce a measure for how far a type space is from having a common prior, which we term prior distance. If a type
More informationOnline Appendix Feedback Effects, Asymmetric Trading, and the Limits to Arbitrage
Online Appendix Feedback Effects, Asymmetric Trading, and the Limits to Arbitrage Alex Edmans LBS, NBER, CEPR, and ECGI Itay Goldstein Wharton Wei Jiang Columbia May 8, 05 A Proofs of Propositions and
More informationA Note on Lobbying a Legislature
TSE 673 July 2016 A Note on Lobbying a Legislature Vera Zaporozhets A Note on Lobbying a Legislature Vera Zaporozhets July 2016 Abstract We study a simple influence game, in which a lobby tries to manipulate
More informationWorking Paper Series
RGEA Universidade de Vigo http://webs.uvigo.es/rgea Working Paper Series A Market Game Approach to Differential Information Economies Guadalupe Fugarolas, Carlos HervésBeloso, Emma Moreno García and
More information17.6.1 Introduction to Auction Design
CS787: Advanced Algorithms Topic: Sponsored Search Auction Design Presenter(s): Nilay, Srikrishna, Taedong 17.6.1 Introduction to Auction Design The Internet, which started of as a research project in
More informationTHE ROOMMATES PROBLEM DISCUSSED
THE ROOMMATES PROBLEM DISCUSSED NATHAN SCHULZ Abstract. The stable roommates problem as originally posed by Gale and Shapley [1] in 1962 involves a single set of even cardinality 2n, each member of which
More informationCHAPTER 3 Numbers and Numeral Systems
CHAPTER 3 Numbers and Numeral Systems Numbers play an important role in almost all areas of mathematics, not least in calculus. Virtually all calculus books contain a thorough description of the natural,
More information[Refer Slide Time: 05:10]
Principles of Programming Languages Prof: S. Arun Kumar Department of Computer Science and Engineering Indian Institute of Technology Delhi Lecture no 7 Lecture Title: Syntactic Classes Welcome to lecture
More informationChoice under Uncertainty
Choice under Uncertainty Part 1: Expected Utility Function, Attitudes towards Risk, Demand for Insurance Slide 1 Choice under Uncertainty We ll analyze the underlying assumptions of expected utility theory
More informationCooperative Games The Shapley value and Weighted Voting. Yair Zick
Cooperative Games The Shapley value and Weighted Voting Yair Zick The Shapley Value Given a player, and a set, the marginal contribution of to is How much does contribute by joining? Given a permutation
More informationThe StudentProject Allocation Problem
The StudentProject Allocation Problem David J. Abraham, Robert W. Irving, and David F. Manlove Department of Computing Science, University of Glasgow, Glasgow G12 8QQ, UK Email: {dabraham,rwi,davidm}@dcs.gla.ac.uk.
More information2DI36 Statistics. 2DI36 Part II (Chapter 7 of MR)
2DI36 Statistics 2DI36 Part II (Chapter 7 of MR) What Have we Done so Far? Last time we introduced the concept of a dataset and seen how we can represent it in various ways But, how did this dataset came
More informationWe never talked directly about the next two questions, but THINK about them they are related to everything we ve talked about during the past week:
ECO 220 Intermediate Microeconomics Professor Mike Rizzo Third COLLECTED Problem Set SOLUTIONS This is an assignment that WILL be collected and graded. Please feel free to talk about the assignment with
More informationInflation. Chapter 8. 8.1 Money Supply and Demand
Chapter 8 Inflation This chapter examines the causes and consequences of inflation. Sections 8.1 and 8.2 relate inflation to money supply and demand. Although the presentation differs somewhat from that
More information11 Ideals. 11.1 Revisiting Z
11 Ideals The presentation here is somewhat different than the text. In particular, the sections do not match up. We have seen issues with the failure of unique factorization already, e.g., Z[ 5] = O Q(
More informationTHE WHE TO PLAY. Teacher s Guide Getting Started. Shereen Khan & Fayad Ali Trinidad and Tobago
Teacher s Guide Getting Started Shereen Khan & Fayad Ali Trinidad and Tobago Purpose In this twoday lesson, students develop different strategies to play a game in order to win. In particular, they will
More information1 Portfolio mean and variance
Copyright c 2005 by Karl Sigman Portfolio mean and variance Here we study the performance of a oneperiod investment X 0 > 0 (dollars) shared among several different assets. Our criterion for measuring
More information6 EXTENDING ALGEBRA. 6.0 Introduction. 6.1 The cubic equation. Objectives
6 EXTENDING ALGEBRA Chapter 6 Extending Algebra Objectives After studying this chapter you should understand techniques whereby equations of cubic degree and higher can be solved; be able to factorise
More informationTangent and normal lines to conics
4.B. Tangent and normal lines to conics Apollonius work on conics includes a study of tangent and normal lines to these curves. The purpose of this document is to relate his approaches to the modern viewpoints
More informationThe Two Envelopes Problem
1 The Two Envelopes Problem Rich Turner and Tom Quilter The Two Envelopes Problem, like its better known cousin, the Monty Hall problem, is seemingly paradoxical if you are not careful with your analysis.
More information6.042/18.062J Mathematics for Computer Science. Expected Value I
6.42/8.62J Mathematics for Computer Science Srini Devadas and Eric Lehman May 3, 25 Lecture otes Expected Value I The expectation or expected value of a random variable is a single number that tells you
More information9. Sampling Distributions
9. Sampling Distributions Prerequisites none A. Introduction B. Sampling Distribution of the Mean C. Sampling Distribution of Difference Between Means D. Sampling Distribution of Pearson's r E. Sampling
More informationThe Binomial Distribution
The Binomial Distribution James H. Steiger November 10, 00 1 Topics for this Module 1. The Binomial Process. The Binomial Random Variable. The Binomial Distribution (a) Computing the Binomial pdf (b) Computing
More information.4 120 +.1 80 +.5 100 = 48 + 8 + 50 = 106.
Chapter 16. Risk and Uncertainty Part A 2009, Kwan Choi Expected Value X i = outcome i, p i = probability of X i EV = pix For instance, suppose a person has an idle fund, $100, for one month, and is considering
More informationThe Trip Scheduling Problem
The Trip Scheduling Problem Claudia Archetti Department of Quantitative Methods, University of Brescia Contrada Santa Chiara 50, 25122 Brescia, Italy Martin Savelsbergh School of Industrial and Systems
More informationStochastic Inventory Control
Chapter 3 Stochastic Inventory Control 1 In this chapter, we consider in much greater details certain dynamic inventory control problems of the type already encountered in section 1.3. In addition to the
More informationInterest Group Coalitions and Information Transmission
Interest Group Coalitions and Information Transmission Kasia Hebda khebda@princeton.edu September 14, 2011 Abstract Though researchers of interest group behavior consistently note that groups often form
More informationLecture notes for Choice Under Uncertainty
Lecture notes for Choice Under Uncertainty 1. Introduction In this lecture we examine the theory of decisionmaking under uncertainty and its application to the demand for insurance. The undergraduate
More informationGerry Hobbs, Department of Statistics, West Virginia University
Decision Trees as a Predictive Modeling Method Gerry Hobbs, Department of Statistics, West Virginia University Abstract Predictive modeling has become an important area of interest in tasks such as credit
More informationThe Ideal Class Group
Chapter 5 The Ideal Class Group We will use Minkowski theory, which belongs to the general area of geometry of numbers, to gain insight into the ideal class group of a number field. We have already mentioned
More informationApplied Algorithm Design Lecture 5
Applied Algorithm Design Lecture 5 Pietro Michiardi Eurecom Pietro Michiardi (Eurecom) Applied Algorithm Design Lecture 5 1 / 86 Approximation Algorithms Pietro Michiardi (Eurecom) Applied Algorithm Design
More informationhttp://www.jstor.org This content downloaded on Tue, 19 Feb 2013 17:28:43 PM All use subject to JSTOR Terms and Conditions
A Significance Test for Time Series Analysis Author(s): W. Allen Wallis and Geoffrey H. Moore Reviewed work(s): Source: Journal of the American Statistical Association, Vol. 36, No. 215 (Sep., 1941), pp.
More informationCombinatorics: The Fine Art of Counting
Combinatorics: The Fine Art of Counting Week 7 Lecture Notes Discrete Probability Continued Note Binomial coefficients are written horizontally. The symbol ~ is used to mean approximately equal. The Bernoulli
More informationSession 8 Smith, Is There A Prima Facie Obligation to Obey the Law?
Session 8 Smith, Is There A Prima Facie Obligation to Obey the Law? Identifying the Question Not: Does the fact that some act is against the law provide us with a reason to believe (i.e. evidence) that
More informationLecture 8 The Subjective Theory of Betting on Theories
Lecture 8 The Subjective Theory of Betting on Theories Patrick Maher Philosophy 517 Spring 2007 Introduction The subjective theory of probability holds that the laws of probability are laws that rational
More informationPrediction Markets, Fair Games and Martingales
Chapter 3 Prediction Markets, Fair Games and Martingales Prediction markets...... are speculative markets created for the purpose of making predictions. The current market prices can then be interpreted
More informationChapter 21: The Discounted Utility Model
Chapter 21: The Discounted Utility Model 21.1: Introduction This is an important chapter in that it introduces, and explores the implications of, an empirically relevant utility function representing intertemporal
More informationCredible Discovery, Settlement, and Negative Expected Value Suits
Credible iscovery, Settlement, and Negative Expected Value Suits Warren F. Schwartz Abraham L. Wickelgren Abstract: This paper introduces the option to conduct discovery into a model of settlement bargaining
More informationCPC/CPA Hybrid Bidding in a Second Price Auction
CPC/CPA Hybrid Bidding in a Second Price Auction Benjamin Edelman Hoan Soo Lee Working Paper 09074 Copyright 2008 by Benjamin Edelman and Hoan Soo Lee Working papers are in draft form. This working paper
More informationEquilibrium: Illustrations
Draft chapter from An introduction to game theory by Martin J. Osborne. Version: 2002/7/23. Martin.Osborne@utoronto.ca http://www.economics.utoronto.ca/osborne Copyright 1995 2002 by Martin J. Osborne.
More informationDiscrete Mathematics and Probability Theory Fall 2009 Satish Rao,David Tse Note 11
CS 70 Discrete Mathematics and Probability Theory Fall 2009 Satish Rao,David Tse Note Conditional Probability A pharmaceutical company is marketing a new test for a certain medical condition. According
More informationInvestigación Operativa. The uniform rule in the division problem
Boletín de Estadística e Investigación Operativa Vol. 27, No. 2, Junio 2011, pp. 102112 Investigación Operativa The uniform rule in the division problem Gustavo Bergantiños Cid Dept. de Estadística e
More informationI d Rather Stay Stupid: The Advantage of Having Low Utility
I d Rather Stay Stupid: The Advantage of Having Low Utility Lior Seeman Department of Computer Science Cornell University lseeman@cs.cornell.edu Abstract Motivated by cost of computation in game theory,
More informationConstructing a TpB Questionnaire: Conceptual and Methodological Considerations
Constructing a TpB Questionnaire: Conceptual and Methodological Considerations September, 2002 (Revised January, 2006) Icek Ajzen Brief Description of the Theory of Planned Behavior According to the theory
More informationStudents in their first advanced mathematics classes are often surprised
CHAPTER 8 Proofs Involving Sets Students in their first advanced mathematics classes are often surprised by the extensive role that sets play and by the fact that most of the proofs they encounter are
More informationChapter 3. Cartesian Products and Relations. 3.1 Cartesian Products
Chapter 3 Cartesian Products and Relations The material in this chapter is the first real encounter with abstraction. Relations are very general thing they are a special type of subset. After introducing
More informationChapter 6: The Information Function 129. CHAPTER 7 Test Calibration
Chapter 6: The Information Function 129 CHAPTER 7 Test Calibration 130 Chapter 7: Test Calibration CHAPTER 7 Test Calibration For didactic purposes, all of the preceding chapters have assumed that the
More informationApproximation Algorithms
Approximation Algorithms or: How I Learned to Stop Worrying and Deal with NPCompleteness Ong Jit Sheng, Jonathan (A0073924B) March, 2012 Overview Key Results (I) General techniques: Greedy algorithms
More informationThe Conference Call Search Problem in Wireless Networks
The Conference Call Search Problem in Wireless Networks Leah Epstein 1, and Asaf Levin 2 1 Department of Mathematics, University of Haifa, 31905 Haifa, Israel. lea@math.haifa.ac.il 2 Department of Statistics,
More informationInformation Theory and Coding Prof. S. N. Merchant Department of Electrical Engineering Indian Institute of Technology, Bombay
Information Theory and Coding Prof. S. N. Merchant Department of Electrical Engineering Indian Institute of Technology, Bombay Lecture  17 ShannonFanoElias Coding and Introduction to Arithmetic Coding
More informationM2L1. Random Events and Probability Concept
M2L1 Random Events and Probability Concept 1. Introduction In this lecture, discussion on various basic properties of random variables and definitions of different terms used in probability theory and
More information1 Introduction to Option Pricing
ESTM 60202: Financial Mathematics Alex Himonas 03 Lecture Notes 1 October 7, 2009 1 Introduction to Option Pricing We begin by defining the needed finance terms. Stock is a certificate of ownership of
More informationPartitioning edgecoloured complete graphs into monochromatic cycles and paths
arxiv:1205.5492v1 [math.co] 24 May 2012 Partitioning edgecoloured complete graphs into monochromatic cycles and paths Alexey Pokrovskiy Departement of Mathematics, London School of Economics and Political
More information2. Information Economics
2. Information Economics In General Equilibrium Theory all agents had full information regarding any variable of interest (prices, commodities, state of nature, cost function, preferences, etc.) In many
More informationEconomics 200B Part 1 UCSD Winter 2015 Prof. R. Starr, Mr. John Rehbeck Final Exam 1
Economics 200B Part 1 UCSD Winter 2015 Prof. R. Starr, Mr. John Rehbeck Final Exam 1 Your Name: SUGGESTED ANSWERS Please answer all questions. Each of the six questions marked with a big number counts
More information