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anigrafs: graphical models and collective choice

based on the research and writings ofWhitman Richards, MIT

presented by Dietrich Falkenthal, dlf@mit.edu

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overview

• tutorial on anigrafs• review of “Collective Choice with Uncertain Domain Models”

and “Graphical Models and Collective Choice”• summary

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vehicles• Valentino Braitenberg wrote Vehicles (1984) to

study how the mind attributes behaviors to the actions of simple machines– simple architecture produces complex behavior– infer cognitive goals from

behaviors (incorrectly)– image on the right shows how

different wiring patterns result in opposite behaviors

– key contribution: modeling of agents interacting with their environment & making choices

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anigrafs

• Anigrafs lay out a simple framework for ways in which cognitive acts may be associated and influence one another– each type of act is associated with an agent along the lines

proposed by Minsky (1986) in Society of Mind

– when in isolation, each agent is mindless– only through conversations with other

agents do mental events and conscious activities emerge

– just as individuals in our society exchange information and cooperate or not, so do anigraf agents communicate and interact, depending upon whether they share similar interests and goals, and have a language in common

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voting procedures

• vote – an individual’s act of expressing preference

• collective choice – group selects one alternative among many

• vote aggregation methods– plurality– Borda count– Condorcet tally– many other procedures to create any desired outcome (D.

Saari, Geometry of Voting)

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anigraf voting example

• pentagon with top-cycle– uses Condorcet tally

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anigrafAn anigraf is a social network of “agents” with these properties:

1. the anigraf is typically depicted as an undirected graph

2. each node, or vertex, in the graph corresponds to an agent (daemon) with a unique first choice preference for the next state of the social system

3. each edge in the graph indicates that there is a common feature or similarity relation between the preferences held by two agents

4. an agent’s ranking of preferences is consistent with the anigraf structure (the form of the graph)

5. each agent has a say in the next state of the social system, with the strength of his vote a variable

6. aggregation of agent's desires will be done using the pair-wise, Condorcet procedure

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anigraf top-cycles

• top-cycles can be produced by varying:– knowledge depth– voting strength– directed or non-directed edges– structure (completeness, bi-partite, etc.)

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anigraf top-cycles• one benefit of a social computation is that behavior can be

changed dramatically, yet predictably, by one vote– example: gaits can be produced by switching between top cycles– notion of achievement of group goal through alignment of preferences

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If group decisions are

(i) based on rational models of choice relationships, and(ii) consensus is reached using a procedure akin to voting,

then how easily can consensus be disrupted?

goal: explore how perturbations in any shared model of the domain will disrupt consensus

Here, we explore the reduced likelihood of unique winners when a shared global model for relating alternatives is violated. The principal result will be that imperfect knowledge of a domain has small consequence if individuals vote faithfully, but haphazard preference orderings that are inconsistent with a shared domain model can create havoc.

graphic modelsand collective choice

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stability & robustness

Stability: for a fixed set of alternatives and model, the stability of an outcome is the probability that there will not be a top-cycle, or, equivalently, that there will be a unique Condorcet winner (excluding ties.)

Robustness: the likelihood that perturbations in the edge set for model, or fluctuations in the weights on alternatives will lead to a different winner

stability measures the ease with which an outcome can be overturned by another alternative, whereas robustness tests whether or not the same outcome will be reached following some perturbation

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random vs. shared preferences

• theoretical results show that for large numbers of voters each having random preferences over a large set of alternatives, there will almost surely be no stable agreement or unique outcome (e.g. Arrow 1963, Campbell & Tullock 1965, Kelly 1986, Saari 1994, Jones et al 1995.)

For the graphical model, shared preferences increases stability

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perturbations in weights vs. edgesThe shared model plays a dominant role in robustness of outcomes

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perturbations in edge directionsRandom assignment of directional edges raises the probability of no

winner by about 5x compared with a random assignment of bi-directional edges.

Consensus favors groups where voters see relationships in the domain in a reciprocal fashion.

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perturbations in rogue votersEven a small number of hap-hazard votes (e.g. 10%) can have severe

consequences on achieving successful outcomes for choice sets larger than 12 alternatives

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results• Richards findings show successful outcomes in collective

decision-making are strongly dependent on the integrity and form of the model of the domain shared by the members of the group.

• Richards expects the results to generalize to other representational forms beyond graphical, as well as to other tally procedures beyond Condorcet, including non-democratic decision-making.

– the results indicate that even for a dictorial regime, a small group of members can create an environment that is potentially unstable, easily reaching a tipping-point

• impacts our understanding of group decision-making, stability of social networks, negotiations seen as collaborations, and collaborations between parties with differing belief structures.

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summary• structure of relationships among the constituents plays an

important role in the knowledge of that society, no matter how small or how large

• this structure, represented as a graph, governs the scope and power of the mental activities and the outcomes chosen by the society as a whole

• the graphical representation is favored to depict the relations between all these varieties of agents simply because this artifact is conceptually very simple, yet has deep and accessible theoretical underpinnings

• because agents or daemons can elicit actions considered “animate acts”, and because they are linked by edges in a larger social network represented as a graph, this society is called an Anigraf

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references[1] W. Richards, B. McKay, and D. Richards. The Probability of Collective Choice with

shared Knowledge Structures. Jrl. Math. Psych. 46, 338-351, 2002.

[2] D. Richards, B. McKay and W. Richards. Collective Choice and Mutual Knowledge Structures. Adv. in Complex Systems 1, 221-236, 1998. Reprinted 2002 Chapt. 17 in Modeling Complexity in Economic & Social Systems. F. Schweitzer (Ed.)

[3] W. Richards and S. Seung, Neural Voting Machines. AI Memo 2004-029, 2004. See also: Ann. Proc. Cog. Sci. Soc. (2003)

[4] W. Richards. Graphical Models and Collective Choice. AFOSR contract #F49620-03-1-0213 (2005).

[5] Valentino Braitenberg. Vehicles: Experiments in Synthethic Psychology, MIT Press, 1984.

[6] D. Saari,. Geometry of Voting. Springer-Verlag, Berlin 1994.

[7] W. Richards, Collective Choice with Uncertain Domain Models, AI Memo 2005-024 16 Aug 05

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backup slides

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voting: plurality procedure

• plurality – the one receiving the most votes wins.

• problems:

– winner may represent a small percentage and still win

– in a pair-wise contest, the outcome frequently will be different (Saari, 1991)

• voters have some information about the set of choices and use this information to vote

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voting: Borda count• Borda count – determines the winner of an election by giving

each candidate a certain number of points corresponding to the position in which he or she is ranked by each voter. Once all votes have been counted the candidate with the most points is the winner.

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voting: Condorcet tally• Condorcet tally – winner is found by conducting a series of

pairwise comparisons pitting every candidate against every other candidate in a series of imaginary one-on-one contests.

• The winner of each pairing is the candidate preferred by a majority of voters. When all possible pairings of candidates have been considered, if one candidate beats every other candidate in these contests then they are declared the Condorcet winner.

– For example, if Alice is paired against Bob it is necessary to count both the number of voters who have ranked Alice higher than Bob, and the number who have ranked Bob higher than Alice. If Alice is preferred by more voters then she is the winner of that pairing.

– In a three candidate pairwise comparison, if Alice beats Bob, Bob beats Gargamel, and Gargamel beats Alice, we have a Condorcet paradox!

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voting: Condorcet paradox

Alice

BobGargamel

Alice Bob Gargamel Alice

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graphic modelsand collective choice

Given a graphical model for the similarity relations between choices, there are 3 obvious ways to affect consensus:

(1) unintentional "noise" distributed among all the voters introduced simply by uncertainty

(2) "rogue" voters who intentionally try to disrupt the process by casting votes arbitrarily

(3) voters whose viewpoints violate the shared model for the domain held by all other voters

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shared models• to insure consensus, some form of constraint must be

introduced that prohibits voters from choosing alternatives haphazardly

• one plausible constraint is that individual preference orders are consistent with a shared global model for relating alternatives (Runkel, 1956)– In this case, the probability of the group reaching a stable

agreement is over 90% (Richards et al., 1998, 2002.)

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shared models• For certain types of shared models, agreement is

guaranteed regardless of the numbers of voters and their voting power. Simple examples of shared models relating alternatives include:– how presidential candidates are positioned along a liberal to

conservative dimension,– the organization of taste choices for soft (or alcoholic!) drinks,– the perceived relation between landmarks in a city,– the democratic versus communist versus industrial wealth of

nations, etc.

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shared models• To insure consensus, important conditions include

(i) that each individual votes faithfully, or not at all when "in doubt",

(ii) that there is no uncertainty or external source of noise that perturbs a voter’s ranking of alternatives, and

(iii) that a voter’s ranking is consistent with the shared global model.

• Any violations of these conditions will reduce the odds for consensus.