A New Outside Option Value for Networks: The Kappa-Value – Measuring Distribution of Power of Political Agreements
In an economic or social situation where agents have to group in order to achieve common goals, how can we calculate the coalitional rents of the agents arising from the coalition formation? Once we have formalized the situation via a TU-game and a network describing the economic structure, we can apply different allocation rules to assign the coalitional rents to the agents. We specifically analyze situations where parties with a specific vote distribution in a parliament have to build agreements in order to reach some required quorum. In this situation, we want to measure the (relative) distribution of power. We analyze the allocation rules called Position value (Meessen (1988) and Borm et al. (1992)) and graph-chi-value (Casajus (2009)). Applying the generalized framework (Gomez et al. (2008)), a framework where coalitions are not established yet, we fi nd that the graph-chi-value does not differ for networks referring to the same coalition while the Position value takes into account the specific role of an agent within the network, i.e. the communication path. We define and characterize a new outside option sensitive value, the Kappa-value, which takes into account both outside options and the role of an agent within the network.
The class of ASN-position values
This paper introduces a class of cooperative allocation rules for TU-games with a network structure that account for an agent’s centrality in the network. In contrast to existing centrality measures, this approach analyzes the consequences of tie failure (rather than node failure). Though not directly applied to the centrality issue, tie failures are the idea underlying the Position value (the Shapley value of the arc-game). In contrast, we allow for a whole class of allocation rules as a basis since the Shapley approach might be unreasonable in political applications, especially in networks with incompatibilities. Requiring additivity (A), equal treatment of equals (Symmetry S), and irrelevance of unproductive agents (nullplayer irrelevance N) for the underlying basis, we define the class of ASN-position values and provide axiomatic characterizations. To emphasize the crucial role of links, we refine this class to multiplicative ASN-position values and provide a monotonicity result with respect to links. We apply our approach to the case of the 2001 state parliament elections in Hamburg, Germany to use our allocation rule as a power index and further provide an example where we use our approach as a centrality measure to identify top key nodes. In both applications, the position value taking the Banzhaf value as a basis turns out to be more convincing than the Shapley-based approach.
Strategy and Manipulation in Medieval Elections
There are many goals in developing electoral protocols, including a desire for a system which is transparent, in that it is clear what the rule or procedure to follow is; nonmanipulable, in that it is not in a person’s best interest to misrepresent their preferences; honest, in the sense that it elects the ‘right’ candidate; and not open to strategizing, i.e., bribery or collusion. However, these desiderata are in tension with each other: Often, transparent electoral procedures are the least strategy resistant, and many honest procedures encourage manipulation. Thus a balance between these different goals must be sought. In modern times, since the seminal result on vote manipulation, the Gibbard-Satterthwaite Theorem [5, 6], much attention has been devoted to developing voting rules where manipulation is never in the best interest of the voters [4] or which are computationally too complex for the average bounded agent to be able to manipulate [1]. This focus on computational aspects of electoral methods is one of the hallmarks of modern studies on voting. But pursuit of these goals is not restricted to modern times: Those participating in elections in the Middle Ages also sought transparency, non-manipulability, honesty, and strategyproofness in so far as these properties can be consistently expressed in a single procedure. However, given the lack of computational sophistication in the Middle Ages, alternate approaches were needed in order to promote honesty, discourage strategizing, etc. These approaches can be classified as either external (constraints introduced outside of the electoral procedure, such as incentives for coming to consensus quickly) or internal (constraints introduced within the electoral procedure, such as voting rules which cannot be manipulated without adverse effects, or which are too difficult for the average bounded agent to manipulate). Surveying examples of both approaches in the context of medieval ecclesiastical and secular elections provides an interesting comparison to modern electoral procedures. Elections in the Middle Ages were used for the same reasons that they are today: To select suitable candidate(s) for a particular office, duty, or obligation. However, it is important to note that the term electio was used in the Middle Ages in a broader sense than our modern ‘election’. Its primary sense was ‘selection’ or ‘choice’, and only secondarily ‘election’ in the modern sense. Thus, many records which purportedly discuss elections are not discussing elections of the type which interests us. We can identify four categories of medieval electoral processes: (1) Election by an external authority having no direct interest in the election; (2) Indirect election, where electors name other electors who then select or elect the officials; (3) Election by lot; and (4) Election by ballot. Elections of the first and third types are generally computationally uninteresting; the first type corresponds to dictatorial voting rules, and the third type collapses to probability theory. In general, interesting voting methods are found only in the fourth type, election by ballot, though they can also occur in indirect election.