Matrix Representations and Extension of the Graph Model for Conflict Resolution

The graph model for conflict resolution (GMCR) provides a convenient and effective means to model and analyze a strategic conflict. Standard practice is to carry out a stability analysis of a graph model, and then to follow up with a post-stability analysis, two critical components of which are status quo analysis and coalition analysis. In stability analysis, an equilibrium is a state that is stable for all decision makers (DMs) under appropriate stability definitions or solution concepts. Status quo analysis aims to determine whether a particular equilibrium is reachable from a status quo (or an initial state) and, if so, how to reach it. A coalition is any subset of a set of DMs. The coalition stability analysis within the graph model is focused on the status quo states that are equilibria and assesses whether states that are stable from individual viewpoints may be unstable for coalitions. Stability analysis began within a simple preference structure which includes a relative preference relationship and an indifference relation. Subsequently, preference uncertainty and strength of preference were introduced into GMCR but not formally integrated. In this thesis, two new preference frameworks, hybrid preference and multiple-level preference, and an integrated algebraic approach are developed for GMCR. Hybrid preference extends existing preference structures to combine preference uncertainty and strength of preference into GMCR. A multiple-level preference framework expands GMCR to handle a more general and flexible structure than any existing system representing strength of preference. An integrated algebraic approach reveals a link among traditional stability analysis, status quo analysis, and coalition stability analysis by using matrix representation of the graph model for conflict resolution. To integrate the three existing preference structures into a hybrid system, a new preference framework is proposed for graph models using a quadruple relation to express strong or mild preference of one state or scenario over another, equal preference, and an uncertain preference. In addition, a multiple-level preference framework is introduced into the graph model methodology to handle multiple-level preference information, which lies between relative and cardinal preferences in information content. The existing structure with strength of preference takes into account that if a state is stable, it may be either strongly stable or weakly stable in the context of three levels of strength. However, the three-level structure is limited in its ability to depict the intensity of relative preference. In this research, four basic solution concepts consisting of Nash stability, general metarationality, symmetric metarationality, and sequential stability, are defined at each level of preference for the graph model with the extended multiple-level preference. The development of the two new preference frameworks expands the realm of applicability of the graph model and provides new insights into strategic conflicts so that more practical and complicated problems can be analyzed at greater depth. Because a graph model of a conflict consists of several interrelated graphs, it is natural to ask whether well-known results of Algebraic Graph Theory can help analyze a graph model. Analysis of a graph model involves searching paths in a graph but an important restriction of a graph model is that no DM can move twice in succession along any path. (If a DM can move consecutively, then this DM's graph is effectively transitive. Prohibiting consecutive moves thus allows for graph models with intransitive graphs, which are sometimes useful in practice.) Therefore, a graph model must be treated as an edge-weighted, colored multidigraph in which each arc represents a legal unilateral move and distinct colors refer to different DMs. The weight of an arc could represent some preference attribute. Tracing the evolution of a conflict in status quo analysis is converted to searching all colored paths from a status quo to a particular outcome in an edge-weighted, colored multidigraph. Generally, an adjacency matrix can determine a simple digraph and all state-by-state paths between any two vertices. However, if a graph model contains multiple arcs between the same two states controlled by different DMs, the adjacency matrix would be unable to track all aspects of conflict evolution from the status quo. To bridge the gap, a conversion function using the matrix representation is designed to transform the original problem of searching edge-weighted, colored paths in a colored multidigraph to a standard problem of finding paths in a simple digraph with no color constraints. As well, several unexpected and useful links among status quo analysis, stability analysis, and coalition analysis are revealed using the conversion function. The key input of stability analysis is the reachable list of a DM, or a coalition, by a legal move (in one step) or by a legal sequence of unilateral moves, from a status quo in 2-DM or $n$-DM ($n > 2$) models. A weighted reachability matrix for a DM or a coalition along weighted colored paths is designed to construct the reachable list using the aforementioned conversion function. The weight of each edge in a graph model is defined according to the preference structure, for example, simple preference, preference with uncertainty, or preference with strength. Furthermore, a graph model and the four basic graph model solution concepts are formulated explicitly using the weighted reachability matrix for the three preference structures. The explicit matrix representation for conflict resolution (MRCR) that facilitates stability calculations in both 2-DM and $n$-DM ($n > 2$) models for three existing preference structures. In addition, the weighted reachability matrix by a coalition is used to produce matrix representation of coalition stabilities in multiple-decision-maker conflicts for the three preference frameworks. Previously, solution concepts in the graph model were traditionally defined logically, in terms of the underlying graphs and preference relations. When status quo analysis algorithms were developed, this line of thinking was retained and pseudo-codes were developed following a similar logical structure. However, as was noted in the development of the decision support system (DSS) GMCR II, the nature of logical representations makes coding difficult. The DSS GMCR II, is available for basic stability analysis and status quo analysis within simple preference, but is difficult to modify or adapt to other preference structures. Compared with existing graphical or logical representation, matrix representation for conflict resolution (MRCR) is more effective and convenient for computer implementation and for adapting to new analysis techniques. Moreover, due to an inherent link between stability analysis and post-stability analysis presented, the proposed algebraic approach establishes an integrated paradigm of matrix representation for the graph model for conflict resolution

A matrix-based approach to searching colored paths in a weighted colored multidigraph

An algebraic approach to finding all edge-weighted-colored paths within a weighted colored multidigraph is developed. Generally, the adjacency matrix represents a simple digraph and determines all paths between any two vertices, and is not readily extendable to colored multidigraphs. To bridge the gap, a conversion function is proposed to transform the original problem of searching edge-colored paths in a colored multidigraph to a standard problem of finding paths in a simple digraph. Moreover, edge weights can be used to represent some preference attribute. Its potentially wide realm of applicability is illustrated by a case study: status quo analysis in the graph model for conflict resolution. The explicit matrix function is more convenient than other graphical representations for computer implementation and for adapting to other applications. Additionally, the algebraic approach reveals the relationship between a colored multidigraph and a simple digraph, thereby providing new insights into algebraic graph theory

A matrix approach to status quo analysis in the graph model for conflict resolution

An algebraic method is developed to carry out status quo analysis within the framework of the graph model for conflict resolution. As a form of post-stability analysis, status quo analysis aims at confirming that possible equilibria, or states stable for all decision-makers, are in fact reachable from the status quo or any other initial state. Although pseudo-codes for status quo analysis have been developed, they have never been implemented within a practical decision support system. The novel matrix approach to status quo analysis designed here is convenient for computer implementation and easy to employ, as is illustrated by an application to a real-world conflict case. Moveover, the proposed explicit matrix approach reveals an inherent link between status quo analysis and the traditional stability analysis and, hence, provides the possibility of establishing an integrated paradigm for stability and status quo analyses

Dynamics of the Presidential Veto: A Computational

We specify and compute equilibria of a dynamic policy-making game between a president and a legislature under insitutional rules that emulate those of the US Constitution. Policies are assumed to lie in a two-dimensional space in which one issue dimension captures systemic differences in partisan preferences, while the other summarizes non-partisan attributes of policy. In any period, the policy choices of politicians are influenced by the position of the status quo policy in this space, with the current policy outcome determining the location of the status quo in the next period. Partisan control of the legislature and presidency changes probabilistically over time. We find that politicians strategically compromise their ideal policy in equilibrium, and that the degree of compromise increases when the opposition party is more likely to take control of the legislature in the next period, while politicians become relatively more extreme when the opposition party is more likely to control the presidency. We measure gridlock by (the inverse of ) the expected distance of enacted policies from the status quo in the long run, and we show that both gridlock and the long run welfare of a representative voter are maximized when government is divided without a super majority in the legislature. Under unified government, we find that the endogeneity of the status quo leads to a non-monotonic effect of the size of the legislative ma jority on gridlock; surprisingly, under unified government, gridlock is higher when the party in control of the legislature has a superma jority than when it has a bare ma jority. Furthermore, a relatively larger component of policy change occurs in the non-partisan policy dimension when a superma jority controls the legislature. We conduct constitutional experiments, and we find that voter welfare is minimized when the veto override provision is abolished and maximized when the presidential veto is abolished.

Determining the parameters in a social welfare function using stated preference data: an application to health

One way in which economists might determine how best to balance the competing objectives of efficiency and equity is to specify a social welfare function (SWF). This paper looks at how the stated preferences of a sample of the general public can be used to estimate the shape of the SWF in the domain of health benefits. The results suggest that it is possible to determine the parameters in a social welfare function from stated preference data, but show that people are sensitive to what inequalities exist and to the groups across which those inequalities exist

The Second-Order Impact of Relative Power on Outcomes of Crisis Bargaining: A Theory of Expected Disutility and Resolve

How does structure shape behavior and outcomes in crisis bargaining? Formal bargaining models of war rely on expected utility theory to describe first-order effects, whereby the payoffs of war determine actors’ “resolve” to fight as a function of costs and benefits. Value preferences of risk and future discounting are routinely treated as predefined and subjective individual attributes, outside the strategic context of bargaining or independent from expected utility. However, such treatment fails to account for context-conditional preferences sourcing from actors’ expectations of relative gain or loss. Drawing on a wealth of experimental evidence from behavioral economics, but without departing from rational choice or compromising theoretical parsimony, this dissertation proposes a systematic differentiation of value preferences conditional on anticipated gain/loss, i.e., the endogenous shift in power bargaining is expected to produce. Whereas the utility of gain incentivizes a challenge to the status quo, the disutility of loss imposes reactive resolve via asymmetrical risk-acceptance and lower discounting of future payoffs. The proposed theory of reactive resolve, thus, reveals the second-order impact of structural conditions on behavior and outcomes in crisis bargaining. Short of this behavioral effect, bargaining models exhibit a tendency of automatic adjustment of benefits which fails to capture the very essence of conflict and encourages erroneous hypotheses about the role of superiority, such as the nuclear superiority hypothesis reviewed and rejected as part of this research. The prescriptive and predictive inaccuracy of the standard rationalist approach is evident in the solution of the most fundamental bargaining problem - a credible commitment problem arising in the context of “bargaining over future bargaining power” (Fearon 1996). By formally integrating and simulating expected gain- and loss-induced preferences, this study demonstrates substantial deviations from previous results. Based on the findings, several theoretical and empirical implications are derived concerning the mechanism of crisis escalation, the relationship between the distribution of power and the likelihood of war, and the challenge to coercion. The prescribed mechanism is then empirically tested against cases of compellence and deterrence, including two of the most significant cases of nuclear crisis, using process tracing as a qualitative tool of causal inference

A Newton Collocation Method for Solving Dynamic Bargaining Games

We develop and implement a collocation method to solve for an equilibrium in the dynamic legislative bargaining game of Duggan and Kalandrakis (2008). We formulate the collocation equations in a quasi-discrete version of the model, and we show that the collocation equations are locally Lipchitz continuous and directionally differentiable. In numerical experiments, we successfully implement a globally convergent variant of Broyden's method on a preconditioned version of the collocation equations, and the method economizes on computation cost by more than 50% compared to the value iteration method. We rely on a continuity property of the equilibrium set to obtain increasingly precise approximations of solutions to the continuum model. We showcase these techniques with an illustration of the dynamic core convergence theorem of Duggan and Kalandrakis (2008) in a nine-player, two-dimensional model with negative quadratic preferences.

Simple Heuristics in Complex Networks: Models of Social Influence

The concept of heuristic decision making is adapted to dynamic influence processes in social networks. We report results of a set of simulations, in which we systematically varied: a) the agents\' strategies for contacting fellow group members and integrating collected information, and (b) features of their social environment—the distribution of members\' status, and the degree of clustering in their network. As major outcome variables, we measured the speed with which the process settled, the distributions of agents\' final preferences, and the rate with which high-status members changed their initial preferences. The impact of the agents\' decision strategies on the dynamics and outcomes of the influence process depended on features of their social environment. This held in particular true when agents contacted all of the neighbors with whom they were connected. When agents focused on high-status members and did not contact low-status neighbors, the process typically settled more quickly, yielded larger majority factions and fewer preference changes. A case study exemplifies the empirical application of the model.Decision Making; Cognition; Heuristics; Small World Networks; Social Influence; Bounded Rationality

The Evolution of Our Preferences: Evidence from Capuchin-Monkey Trading Behavior

Behavioral economics has demonstrated systematic decision-making biases in both lab and field data. But are these biases learned or innate? We investigate this question using experiments on a novel set of subjects — capuchin monkeys. By introducing a fiat currency and trade to a capuchin colony, we are able to recover their preferences over a wide range of goods and risky choices. We show that standard price theory does a remarkably good job of describing capuchin purchasing behavior; capuchin monkeys react rationally to both price and wealth shocks. However, when capuchins are faced with more complex choices including risky gambles, they display many of the hallmark biases of human behavior, including reference-dependent choices and loss-aversion. Given that capuchins demonstrate little to no social learning and lack experience with abstract gambles, these results suggest that certain biases such as loss-aversion are an innate function of how our brains code experiences, rather than learned behavior or the result of misapplied heuristics.Prospect theory, Loss aversion, Reference dependence, Evolution, Neuroeconomics, Capuchin monkeys, Monkey business