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Stable outcomes in spatial voting games
A
tion,
and a probability
observations.) IS have a unique
model (a joint
property
213
bstrac,ts
probability
distribution
which makes them especially
on hypotheses
suitable
as elements
and for
a logic; namely, they permit dominance. One IS can have a higher expected value than another for all payoff functions. A well-known example of this property is the positive value of information princple - additional information is always valuable (or at least, never harmful). The dominance relation is a partial order - transitive, reflexive and anti-symmetric. For IS with binary hypotheses (IS which address a yes-no question), dominance is a lattice (every pair of IS have a least upper bound and a greatest lower bound.) It is only a short step from the lattice to a logical calculus. A set of postulates for the calculus has been formulated and will be presented, as well as some of the more salient theorems. Among the latter is the interesting rule of reduction which states that for three information systems, even though they are mutually non-dominating (each contains information not contained in the other two), there are conditions in which one can be ignored. The logic has implications for the design of information systems, e.g., the construction of expert systems in artificial intelligence, and for the aggregation of information in decisions under uncertainty. A simple medical example will be presented to illustrate these potentialities.
Stable
Outcomes
in
Spatial
Voting
Games.
Guiflermo
Owen,
Deparfment
of
Mathematics, Naval Postgraduate School, Monterey, CA 93943, U.S.A., and School of Social Sciences, University of California, Irvine, CA 92717, U.S.A.
The traditional representation of outcomes for an n-person game as vectors in n-space was superseded in the 1960s when it became clear that, for political and related applications, a lower-dimensional space was generally sufficient to the purpose. Thus, instead of vectors in utility space, vectors in policy space came to be used. Utility to the various decision-makers (usually voters or parliamentarians) could then be defined in terms of distance from given ideal points. A well-known result states that, when the underlying policy space is one-dimensional, a non-empty core (one or more undominated points) will exist. In two or more dimensions, it is known that, for decisive games, where a bare majority is sufficient to carry motions, the core will be non-empty only under very special circumstances (degenerate cases). Stability is then sought, in the form of near-core outcomes. Among the best-known of the near-core outcomes are (a) the Copeiand winner, defined as that outcome which defeats the greatest proportion of alternatives; (b) the yolk center, defined as that alternative which comes closest to satisfying all possible winning coalitions of voters; and (c) the minimal response (finagle) point, defined as the point which comes closest to defeating any other alternative.
21-l
A bsrrucrs
iL.c show that each of these near-core outcomes has a well defined analogue for games vvith side payments (classical von Neumann-Morgenstern games). In particular, the Copeland winner’s analogue is the Shapley value and the yolk center’s analogue is the nucleolus. The side-payment analogue of the finagle point has not, hovvever, been previously studied. We derive some of its properties and show how it may be easily obtained.
Interactive Research,
Strategic
Graduate
Analysis.
Selwyn
Enzer,
School of Business Administrution.
USC.
International Los Angeles,
Business CA 50085,
Education
and
U.S.A.
Poor strategic performance in public and prive organizations is often the result of unexpected changes in the external environment - planning for one scenario and having another emerge. Strategic planning is supposed to help organizations cope with change, and it does help when the planning assumptions closely match the external environment. However, most strategic planning procedures are limited to a narrow set of assumptions about future change and are unable to deal effectively with even a small number of radically different scenarios. What many organizations seek is a ‘robust’ strategy, one that can accommodate or adapt to many possible scenarios, as opposed to one that may be optimal under the ‘most likely scenario’ but difficult to manage in another scenario. The questions are, how to identify strategically significant alternative scenarios and how to design a robust strategy? An interactive strategic analysis can help in these efforts. As pointed out by Peter Drucker, George Steiner and others, strategic planning is not concerned with what the organization should do tomorrow, but what it should do today to be ready for an uncertain future. Yet these same authors also point out that most strategic planning processes concentrate on ‘most likely future conditions’, with alternative scenarios - the major source of strategic uncertainty - being relegated to the impotent role of contingency planning. Indeed, most strategic planning procedures eliminate most alternative scenarios by requiring that executive management set the strategic assumptions, objectives, and policies, based on their subjective beliefs (or hopes), prior to the initiation of the planning process itself. An interactive strategic analysis is an open-ended computer simulation that is intended to help executive management in this task by analyzing a broad range of alternative scenarios, before deciding on the strategic assumptions, goals, and range of policies that should be used in the planning process. Unfortunately, very little analytic research has been devoted to the development of methods to explore open-ended problems such as are encountered in the study of alternative scenarios. INTERAX, a man-machine computer simulation developed by the author, is a preliminary version of an interactive strategic analysis that brings together the strengths of expert opinion, corporate simulation models, cross-impact models, and human judgment in a process that can generate a broad range of alternative scenarios and the accompanying ‘if-then’ organizational forecasts. INTERAX com-
tion,
and a probability
observations.) IS have a unique
model (a joint
property
213
bstrac,ts
probability
distribution
which makes them especially
on hypotheses
suitable
as elements
and for
a logic; namely, they permit dominance. One IS can have a higher expected value than another for all payoff functions. A well-known example of this property is the positive value of information princple - additional information is always valuable (or at least, never harmful). The dominance relation is a partial order - transitive, reflexive and anti-symmetric. For IS with binary hypotheses (IS which address a yes-no question), dominance is a lattice (every pair of IS have a least upper bound and a greatest lower bound.) It is only a short step from the lattice to a logical calculus. A set of postulates for the calculus has been formulated and will be presented, as well as some of the more salient theorems. Among the latter is the interesting rule of reduction which states that for three information systems, even though they are mutually non-dominating (each contains information not contained in the other two), there are conditions in which one can be ignored. The logic has implications for the design of information systems, e.g., the construction of expert systems in artificial intelligence, and for the aggregation of information in decisions under uncertainty. A simple medical example will be presented to illustrate these potentialities.
Stable
Outcomes
in
Spatial
Voting
Games.
Guiflermo
Owen,
Deparfment
of
Mathematics, Naval Postgraduate School, Monterey, CA 93943, U.S.A., and School of Social Sciences, University of California, Irvine, CA 92717, U.S.A.
The traditional representation of outcomes for an n-person game as vectors in n-space was superseded in the 1960s when it became clear that, for political and related applications, a lower-dimensional space was generally sufficient to the purpose. Thus, instead of vectors in utility space, vectors in policy space came to be used. Utility to the various decision-makers (usually voters or parliamentarians) could then be defined in terms of distance from given ideal points. A well-known result states that, when the underlying policy space is one-dimensional, a non-empty core (one or more undominated points) will exist. In two or more dimensions, it is known that, for decisive games, where a bare majority is sufficient to carry motions, the core will be non-empty only under very special circumstances (degenerate cases). Stability is then sought, in the form of near-core outcomes. Among the best-known of the near-core outcomes are (a) the Copeiand winner, defined as that outcome which defeats the greatest proportion of alternatives; (b) the yolk center, defined as that alternative which comes closest to satisfying all possible winning coalitions of voters; and (c) the minimal response (finagle) point, defined as the point which comes closest to defeating any other alternative.
21-l
A bsrrucrs
iL.c show that each of these near-core outcomes has a well defined analogue for games vvith side payments (classical von Neumann-Morgenstern games). In particular, the Copeland winner’s analogue is the Shapley value and the yolk center’s analogue is the nucleolus. The side-payment analogue of the finagle point has not, hovvever, been previously studied. We derive some of its properties and show how it may be easily obtained.
Interactive Research,
Strategic
Graduate
Analysis.
Selwyn
Enzer,
School of Business Administrution.
USC.
International Los Angeles,
Business CA 50085,
Education
and
U.S.A.
Poor strategic performance in public and prive organizations is often the result of unexpected changes in the external environment - planning for one scenario and having another emerge. Strategic planning is supposed to help organizations cope with change, and it does help when the planning assumptions closely match the external environment. However, most strategic planning procedures are limited to a narrow set of assumptions about future change and are unable to deal effectively with even a small number of radically different scenarios. What many organizations seek is a ‘robust’ strategy, one that can accommodate or adapt to many possible scenarios, as opposed to one that may be optimal under the ‘most likely scenario’ but difficult to manage in another scenario. The questions are, how to identify strategically significant alternative scenarios and how to design a robust strategy? An interactive strategic analysis can help in these efforts. As pointed out by Peter Drucker, George Steiner and others, strategic planning is not concerned with what the organization should do tomorrow, but what it should do today to be ready for an uncertain future. Yet these same authors also point out that most strategic planning processes concentrate on ‘most likely future conditions’, with alternative scenarios - the major source of strategic uncertainty - being relegated to the impotent role of contingency planning. Indeed, most strategic planning procedures eliminate most alternative scenarios by requiring that executive management set the strategic assumptions, objectives, and policies, based on their subjective beliefs (or hopes), prior to the initiation of the planning process itself. An interactive strategic analysis is an open-ended computer simulation that is intended to help executive management in this task by analyzing a broad range of alternative scenarios, before deciding on the strategic assumptions, goals, and range of policies that should be used in the planning process. Unfortunately, very little analytic research has been devoted to the development of methods to explore open-ended problems such as are encountered in the study of alternative scenarios. INTERAX, a man-machine computer simulation developed by the author, is a preliminary version of an interactive strategic analysis that brings together the strengths of expert opinion, corporate simulation models, cross-impact models, and human judgment in a process that can generate a broad range of alternative scenarios and the accompanying ‘if-then’ organizational forecasts. INTERAX com-