Territorial Division: . . . Nash’s Solution: From . . . Nash’s Solution: . . . Nash’s Solution: . . . How to Make a Solution to How to Take . . . a Territorial Dispute More How to Take Emotions . . . What If Emotions Are . . . Realistic: Taking into Immediate Solution . . . Auxiliary Question: . . . Account Uncertainty, Home Page Emotions, and Title Page Step-by-Step Approach ◭◭ ◮◮ ◭ ◮ Mahdokht Afravi and Vladik Kreinovich Page 1 of 16 Department of Computer Science University of Texas at El Paso Go Back El Paso, TX 79968, USA Full Screen mafravi@miners.utep.edu, vladik@utep.edu Close Quit
Territorial Division: . . . Nash’s Solution: From . . . 1. Territorial Division: Formulation of the Prob- Nash’s Solution: . . . lem Nash’s Solution: . . . • In many real-life situations, there is a dispute over a How to Take . . . territory: How to Take Emotions . . . What If Emotions Are . . . – from conflicts between neighboring farms Immediate Solution . . . – to conflict between states. Auxiliary Question: . . . • As a result of a conflict, none of the sides can use this Home Page territory efficiently. Title Page • In such situations, it is desirable to come up with a ◭◭ ◮◮ mutually beneficial agreement. ◭ ◮ • The current solution is based on the work by the No- Page 2 of 16 belist J. Nash. Go Back • Nash showed that the best mutually beneficial solution Full Screen maximizes the product of all the utilities. Close Quit
Territorial Division: . . . Nash’s Solution: From . . . 2. Nash’s Solution: From a Theoretical Formula- Nash’s Solution: . . . tion to Practical Recommendations Nash’s Solution: . . . • Let u i ( x ) be the utility (per area) of the i -th participant How to Take . . . at location x . How to Take Emotions . . . What If Emotions Are . . . • We should select a partition for which the product � n Immediate Solution . . . U i is the largest, where: Auxiliary Question: . . . i =1 � Home Page def • U i = S i u i ( x ) dx and Title Page • S i is the set allocated to the i -th participant. ◭◭ ◮◮ • Solution: for some t i , to assign each location x to the ◭ ◮ participant i with the largest ratio u i ( x ) /t i . Page 3 of 16 • The parameters t i must be determined from the re- � n Go Back quirement that the U i → max. i =1 Full Screen • For two participants, x ∈ S 1 if u 1 ( x ) = t 1 def u 2 ( x ) ≥ t . Close t 2 Quit
Territorial Division: . . . Nash’s Solution: From . . . 3. Nash’s Solution: Advantages and Limitations Nash’s Solution: . . . • Nash’s solution is in perfect agreement with common Nash’s Solution: . . . sense description as formalized by fuzzy logic: How to Take . . . How to Take Emotions . . . – we want he first participant to be happy and the What If Emotions Are . . . second participant to be happy, etc. Immediate Solution . . . – the degree of happiness of each participant can be Auxiliary Question: . . . described by his or her utility; Home Page – to represent “and”, it’s reasonable to use one of the Title Page most frequently used fuzzy “and”-operations a · b . ◭◭ ◮◮ • Nash’s solution assumes that we know the exact values u i ( x ). ◭ ◮ • In reality, we know the values u i ( x ) only approximately. Page 4 of 16 • For example, we only know the interval [ u i ( x ) , u i ( x )] Go Back containing u i ( x ). Full Screen • How can we take this uncertainty into account? Close Quit
Territorial Division: . . . Nash’s Solution: From . . . 4. Nash’s Solution: Limitations (cont-d) Nash’s Solution: . . . • The above solution assumes that all the sides are mak- Nash’s Solution: . . . ing their decisions on a purely rational basis. How to Take . . . How to Take Emotions . . . • In reality, emotions are often involved. What If Emotions Are . . . • How can we take these emotions into account? Immediate Solution . . . • Finally, the above formula proposes an immediate so- Auxiliary Question: . . . Home Page lution. Title Page • But participants often follow step-by-step approach: ◭◭ ◮◮ – they first divide a small part, ◭ ◮ – then another part, etc. Page 5 of 16 • This also needs to be taken into account. Go Back • In this talk, we show how to take all this into account. Full Screen Close Quit
Territorial Division: . . . Nash’s Solution: From . . . 5. How to Take Uncertainty into Account Nash’s Solution: . . . • Reminder: we often only know the bounds on u i ( x ): Nash’s Solution: . . . u i ( x ) ≤ u i ( x ) ≤ u i ( x ). How to Take . . . How to Take Emotions . . . • In this case, for each allocation S i , we only know the What If Emotions Are . . . interval [ U i , U i ] of possible values of utility: � � Immediate Solution . . . def def U i = u i ( x ) dx ; U i = u i ( x ) dx Auxiliary Question: . . . S i S i Home Page • In situations with interval uncertainty, decision theory Title Page recommends using � U i = α i · U i + (1 − α i ) · U i ◭◭ ◮◮ • α i ∈ [0 , 1] be i -th participant’s degree of optimism ◭ ◮ � • Similarly, we can use � U i = S i � u i dx, where Page 6 of 16 def � u i ( x ) = α i · u i ( x ) + (1 − α i ) · u i ( x ) Go Back • We acquire the same formulation, so, we assign each lo- Full Screen cation x to a participant with the largest ratio � u i ( x ) /t i . Close Quit
Territorial Division: . . . Nash’s Solution: From . . . 6. Example Nash’s Solution: . . . • Let us assume that different participants assign the Nash’s Solution: . . . same utility to all the locations: How to Take . . . How to Take Emotions . . . u i ( x ) = u j ( x ) and u i ( x ) = u j ( x ) for all i and j. What If Emotions Are . . . • The only difference between the participants is that Immediate Solution . . . they have different optimism degrees α i � = α j . Auxiliary Question: . . . Home Page • Without losing generality, let α i > α j . Title Page • Then, the above optimization implies that a point is ◭◭ ◮◮ allocated to i -th zone if u ( x ) − u ( x ) ≥ t ; so: u ( x ) ◭ ◮ – a more optimistic participant gets the locations Page 7 of 16 with higher uncertainty, while Go Back – a more pessimistic one get locations with lower un- Full Screen certainty. Close Quit
Territorial Division: . . . Nash’s Solution: From . . . 7. How to Take Emotions Into Account Nash’s Solution: . . . • Emotions mean that instead of maximizing U i , partic- Nash’s Solution: . . . = U i + � ipants maximize U emo α ij · U j . How to Take . . . i j How to Take Emotions . . . • Here, α ij describes the feelings of the i -th participant What If Emotions Are . . . towards the j -th one: Immediate Solution . . . • α ij > 0 indicate positive feelings; Auxiliary Question: . . . Home Page • α ij < 0 indicate negative feelings; • α ij = 0 indicate indifference. Title Page • Nash’s solution is to maximize the product � U emo . ◭◭ ◮◮ i i ◭ ◮ • Result: for some t i we assign each location x to a par- Page 8 of 16 ticipant with the largest ratio � u i ( x ) /t i . Go Back • Main difference: the thresholds t i change. Full Screen • A participant with α ij > 0 gets fewer locations x , since his utility is improved via happiness of others. Close Quit
Territorial Division: . . . Nash’s Solution: From . . . 8. What If Emotions Are Negative? Nash’s Solution: . . . • When emotions are negative, i.e., when α ij < 0, then, Nash’s Solution: . . . somewhat surprisingly, we get a positive effect. How to Take . . . How to Take Emotions . . . • Specifically, negative emotions stimulate equality. What If Emotions Are . . . • Indeed, all the sides agree to a division only if their Immediate Solution . . . utilities U emo are non-negative. i Auxiliary Question: . . . • For example, when α 12 = α 21 = − 1, then: Home Page – the only way to guarantee that both values U emo = Title Page 1 U 1 − U 2 and U emo = U 2 − U 1 are non-negative is 2 ◭◭ ◮◮ – when the values U 1 and U 2 are equal to each other. ◭ ◮ • For other values α ij : Page 9 of 16 – we do not get U i = U j , but Go Back – we get bounds limiting how much U i and U j can differ from each other: 0 < c ≤ U i Full Screen ≤ C . U j Close Quit
Territorial Division: . . . Nash’s Solution: From . . . 9. Immediate Solution vs. Step-by-Step Approach Nash’s Solution: . . . • It is desirable to arrive at an immediate solution, but Nash’s Solution: . . . in international affairs, this is not common. How to Take . . . How to Take Emotions . . . • So, we approach the problem in a location-by-location What If Emotions Are . . . basis. Immediate Solution . . . • It turns out that the resulting arrangement is not op- Auxiliary Question: . . . timal. Home Page • In small vicinities of each location x , utility functions Title Page u i ( x ) do not change much. ◭◭ ◮◮ • So, we can safely assume that in the vicinity, each util- ◭ ◮ ity function is a constant u i ( x ) = u i . � Page 10 of 16 • Thus, the utility U i = S i u i ( x ) dx is proportional to the area A i of the set S i : U i = u i · A i . Go Back • Then, the optimal division means selecting A i for Full Screen � � � n n n which A i = A and U i = ( u i · A i ) → max. Close i =1 i =1 i =1 Quit
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