Need for Voting and . . . What Information Can . . . Ranking-Based . . . Why Borda Count? Ranking-Based Voting How to Describe . . . Revisited: Maximum Utility-Based Decision . . . How to Make a Group . . . Entropy Approach Case of Transferable . . . Maximum Entropy . . . Leads to Borda Count Home Page (and Its Versions) Title Page ◭◭ ◮◮ Olga Kosheleva 1 , Vladik Kreinovich 1 , ◭ ◮ and Guo Wei 2 1 University of Texas at El Paso Page 1 of 28 El Paso, Texas 79968, USA Go Back olgak@utep.edu, vladik@utep.edu 2 Department of Mathematics and Computer Science Full Screen University of North Carolina at Pembroke Pembroke, North Carolina 28372 USA, guo.wei@uncp.edu Close Quit
Need for Voting and . . . What Information Can . . . 1. Need for Voting and Group Decision Making Ranking-Based . . . • In many real-life situations, we need to make a decision Why Borda Count? that affects many people. How to Describe . . . Utility-Based Decision . . . • Ideally, when making this decision, we should take into How to Make a Group . . . account the preferences of all the affected people. Case of Transferable . . . • This group decision making situation is also known as Maximum Entropy . . . voting . Home Page Title Page ◭◭ ◮◮ ◭ ◮ Page 2 of 28 Go Back Full Screen Close Quit
Need for Voting and . . . What Information Can . . . 2. What Information Can Be Used for Voting Ranking-Based . . . • The simplest – and most widely used – type of voting Why Borda Count? is when each person selects one of the alternatives. How to Describe . . . Utility-Based Decision . . . • After this selection, all we know is how many people How to Make a Group . . . voted for each alternative; clearly: Case of Transferable . . . – the more people vote for a certain alternative, Maximum Entropy . . . – the better is this alternative for the community as Home Page a whole. Title Page • Thus, if this is all the information we have, then: ◭◭ ◮◮ – a natural idea is ◭ ◮ – to select the alternative that gathered the largest Page 3 of 28 number of votes. Go Back • (Another idea is to keep only the alternatives with the Full Screen largest number of votes and vote again.) Close Quit
Need for Voting and . . . What Information Can . . . 3. What Information Can Be Used (cont-d) Ranking-Based . . . • In this scheme, for each person, Why Borda Count? How to Describe . . . – we only take into account one piece of information: Utility-Based Decision . . . – which alternative is preferable to this person. How to Make a Group . . . • To make more adequate decision, it is desirable to use Case of Transferable . . . more information about people’s preferences. Maximum Entropy . . . • An ideal case is when we use full information about Home Page people’s preferences. Title Page • This is ideal but this requires too much elicitation and ◭◭ ◮◮ is, thus, not used in practice. ◭ ◮ • An intermediate stage – when we use more information Page 4 of 28 than in the simple majority voting – is when: Go Back – we ask the participants to rank all the alternatives, Full Screen and – we use these rankings to make a decision. Close Quit
Need for Voting and . . . What Information Can . . . 4. Ranking-Based Voting: A Brief Reminder Ranking-Based . . . • The famous result by a Nobelist Kenneth Arrow shows: Why Borda Count? How to Describe . . . – that it is not possible to have a ranking-based vot- Utility-Based Decision . . . ing scheme How to Make a Group . . . – which would satisfy all reasonable fairness-related Case of Transferable . . . properties. Maximum Entropy . . . • So what can we do? One of the ideas is Borda count , Home Page when: Title Page – for each participant i and for each alternative A j , ◭◭ ◮◮ – we count the number b ij of alternatives that the ◭ ◮ i -th participant ranked lower than A j . Page 5 of 28 • Then, for each alternative A j , we add up the numbers Go Back corresponding to different participants. Full Screen n � • We select the alternative with the largest sum b ij . Close i =1 Quit
Need for Voting and . . . What Information Can . . . 5. Why Borda Count? Ranking-Based . . . • Borda count is often successfully used in practice. Why Borda Count? How to Describe . . . • However, there are several other alternative schemes. Utility-Based Decision . . . • This prompts a natural question: why namely Borda How to Make a Group . . . count and why not one of these other schemes? Case of Transferable . . . • In this talk, we provide an explanation for the success Maximum Entropy . . . Home Page of Borda count; namely, we show that: Title Page – if we use the maximum entropy approach – a known way for making decisions under uncertainty, ◭◭ ◮◮ – then the Borda count (and its versions) naturally ◭ ◮ follows. Page 6 of 28 Go Back Full Screen Close Quit
Need for Voting and . . . What Information Can . . . 6. How to Describe Individual Preferences Ranking-Based . . . • We want to describe what we should do when only Why Borda Count? know the rankings. How to Describe . . . Utility-Based Decision . . . • Let us first recall what decision we should make when How to Make a Group . . . we have full information about the preferences. Case of Transferable . . . • To describe this, we need to recall how to describe these Maximum Entropy . . . preferences. Home Page • In decision theory, a user’s preferences are described Title Page by using the notion of utility . ◭◭ ◮◮ • To define this notion, we need to select two extreme ◭ ◮ alternatives: Page 7 of 28 – a very bad alternative A − which is worse than any- Go Back thing that we will actually encounter, and Full Screen – a very good alternative A + which is better than anything that we will actually encounter. Close Quit
Need for Voting and . . . What Information Can . . . 7. How to Describe Preferences (cont-d) Ranking-Based . . . • For each number p from the interval [0 , 1], we can then Why Borda Count? form a lottery L ( p ) in which: How to Describe . . . Utility-Based Decision . . . – we get A + with probability p and How to Make a Group . . . – we get A − with the remaining probability 1 − p . Case of Transferable . . . • For p = 0, the lottery L (0) coincides with the very bad Maximum Entropy . . . alternative A − . Home Page • Thus, L (0) is worse than any of the alternatives A that Title Page we encounter: L (0) = A − < A. ◭◭ ◮◮ • For p = 1, the lottery L (1) coincides with the very ◭ ◮ good alternative A + . Page 8 of 28 • Thus, L (1) is better than any of the alternatives A that Go Back we encounter: A < L (1) = A + . Full Screen • Clearly, the larger p , the better the lottery. Close Quit
Need for Voting and . . . What Information Can . . . 8. How to Describe Preferences (cont-d) Ranking-Based . . . • Thus, there exists a threshold p 0 such that: Why Borda Count? How to Describe . . . – for p < p 0 , we have A ( p ) < A , and Utility-Based Decision . . . – for p > p 0 , we have A < A ( p ). How to Make a Group . . . • This threshold is known as the utility of the alterna- Case of Transferable . . . tive A ; it is usually denoted by u ( A ). Maximum Entropy . . . Home Page • In particular, according to this definition: Title Page – the very bad alternative A − has utility 0, while ◭◭ ◮◮ – the very good alternative A + has utility 1. ◭ ◮ • To fully describe people’s preferences, we need to elicit, Page 9 of 28 – from each person i , Go Back – this person’s utility u i ( A j ) of all possible alterna- Full Screen tives A j . Close Quit
Need for Voting and . . . What Information Can . . . 9. Utility Is Defined Modulo Linear Transforma- Ranking-Based . . . tions Why Borda Count? • The numerical value of utility depends on the selection How to Describe . . . of values A − and A + . Utility-Based Decision . . . How to Make a Group . . . • One can show that, if we use a different pair of alter- natives ( A ′ − , A ′ Case of Transferable . . . + ), then: Maximum Entropy . . . – the resulting new utility values u ′ ( A ) are related to Home Page the original values u ( A ) Title Page – by a linear dependence: u ′ ( A ) = k + ℓ · u ( A ) for ◭◭ ◮◮ some k and ℓ > 0. ◭ ◮ Page 10 of 28 Go Back Full Screen Close Quit
Need for Voting and . . . What Information Can . . . 10. Utility-Based Decision Making under Proba- Ranking-Based . . . bilistic Uncertainty Why Borda Count? • In many practical situations, we do not know the exact How to Describe . . . consequences of different actions. Utility-Based Decision . . . How to Make a Group . . . • For each action, we may have different consequences Case of Transferable . . . c 1 , . . . , c m , with different utilities u ( c 1 ) , . . . , u ( c m ). Maximum Entropy . . . • We can also usually estimate the probabilities p 1 , . . . , p m Home Page of different consequences. Title Page • What is the utility of this action? ◭◭ ◮◮ • This action is equivalent to selecting c i with probabil- ◭ ◮ ity p i . Page 11 of 28 • By definition of utility, each consequence c i is, its turn, equivalent to a lottery in which: Go Back Full Screen – we get A + with probability u ( c i ) and – we get A − with the remaining probability 1 − u ( c i ). Close Quit
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