Introduction to Formal Modeling

1. Introduction to Formal Modeling John Aldrich & Arthur Lupia For EITM 2007

2. Outline • The purpose of modeling • Structure • Relevance • Controversy • Elements of Logic

3. Your research design problem • Where are they? • Who is your target audience? • What factual premises/truth claims will they accept. • Where do they want to be? • Which alternate conclusion will benefit them? • What burden of proof and standard of evidence do they impose?

4. Definitions • Theory • Formal Theory • Rational Choice Theory • Rationality • Game Theory • Cooperative • Non-Cooperative*

5. Arguments • The currency of scientific communication. • The components of an argument are: • The Conclusion • The Premises • Value comes from explaining as much as possible with as little as possible.

6. Introduction to Logic • Premise • Conclusion • Logical Validity • Deductive • Inductive • Invalid • Soundness

7. Introduction to Logic • Logical Validity • Deductive • If all of the premises are true, then the conclusion must be true. • The logical connection between premises and conclusion is one of necessity. • Inductive • If all of the premises are true, then the conclusion may be true. • The logical connection between premises and conclusion is one of possibility. • Invalid • If all of the premises are true, then the conclusion must be false. • The logical connection between premises and conclusion is one of impossibility.

8. Examples • George W. Bush is a man. • George W. Bush is over 5’ 11’’ tall. • All men who are over 5’ 11’’ tall are the president. • Therefore, George W. Bush is the president.

9. Examples • George W. Bush is a man. • George W. Bush is over 5’ 11’’ tall. • Some men who are over 5’ 11’’ tall are the president. • Therefore, George W. Bush is the president.

10. Examples • George W. Bush is a man. • George W. Bush is over 5’ 11’’ tall. • If aman is over 5’ 11’’ tall, then he is not the president. • Therefore, George W. Bush is the president.

11. Examples • Almost any random premise 1 • Almost any random premise 2 • Therefore, George W. Bush is the president.

12. The value of logic in debate • How to cast doubt on the reliability of a conclusion when an argument has the following logical properties: • Invalid: • Reveal the logical relationship. • Inductively valid: • show that even if the premises are true the conclusion can be false or demonstrate that one or more of the premises is untrue. • Deductively valid: • demonstrate that one or more of the premises is untrue.

13. Standards for another time • Soundness • Waller (p 20), The argument “must be [deductively] valid and all of its premises must actually be true.” • Reliability • Waller (p 21). “[A]n inductive argument with all true premises, and whose premises strongly support its conclsion ,will be a reliable inductive argument.” • These standards are more subjective.

14. Logical Fallacy: Denying the Antecedent • If it's raining, then the streets are wet. • It isn't raining. • Therefore, the streets aren't wet.

15. Logical Fallacy: Affirming the Consequent • If it's raining then the streets are wet. • The streets are wet. • Therefore, it's raining.

16. Logical Fallacy: Commutation of Conditionals • If James was a bachelor, then he was unmarried. • Therefore, if James was unmarried, then he was a bachelor.

17. The two faces of “or” • Most logic texts claim that "or" has two meanings: • Inclusive (or "weak") disjunction: One or both of the disjuncts is true, which is what is meant by the "and/or" of legalese. • Exclusive (or "strong") disjunction: Exactly one of the disjuncts is true. • Example • Today is Saturday or Sunday. Today is Saturday. Therefore, today is not Sunday • Suppressed premise: Saturday is not Sunday.

18. Logical Fallacy: Denying a Conjunct • It isn't both sunny and overcast. • It isn't sunny. • Therefore, it's overcast. • Not both p and q. • Not p. • Therefore, not q.

19. Ockham’s Razor (14th c.) • Arguments are most helpful to an audience the extent that they actually bring clarity to the phenomena you're studying. • lex parsimoniae entities should not be multiplied beyond necessity • In many cases, less is more.

20. What will you choose? • All political scientists make assumptions about: • Players, Actions, Strategies, Information, Beliefs, Outcomes, Payoffs, and • Method of inference (e.g., “I know it when I see it,” path dependence, Nash Equilibrium, logit plus LLN). • Some state their assumptions more precisely than others. • Conclusions depend on assumptions.

21. Questions to Ask Before You Begin • Is the process static or dynamic? • Do the actors have complete or incomplete information? • Are there many actors or just a few? • How have others modeled the political phenomena you're studying?

22. Starting to write down a formal argument about politics • Make it as simple as possible to start: • Use discrete instead of continuous parameter spaces • Use two actors instead of a zillion • Use complete information rather than incomplete information • Deriving Conclusions • Uniqueness versus existence • Interpreting Your Findings

23. Cooperative Game Theory • Originally dominant, possibly the future?

24. Three Forms of Games • Extensive Form • Normal form • Characteristic Function Form • These differed by level of detail and generality (most detail to most general) • Before the “noncooperative revolution” game theory was based on modeling based on preferences only (not on probabilities a la Harsanyi or on information)

25. Extensive Form • Has the most details of moves • Many of the central results developed in the 1940s and 1950s (including what came to be called sub game perfection and the folk theorem) • It was thought to be practically impossible for studying any realistic setting – too few moves, too few players

26. Normal (Matrix) Form • Became the most used early (in both cooperative and non-cooperative) • Abstracted details of moves (and information sets) to strategies, which seemed like a small loss of detail • A few results on one-to-one mappings between normal form and extensive form equilibrium • Served as the locus for the “debate” between Nash and Von Neumann types.

27. Coalitions and Characteristic Function Form • The perceived weakness of both extensive and normal forms is that each was unmanageable with any realistic number of moves and players • It was also thought to have ignored one of the most important aspects of settings amenable to game theory – coalitions • Political Science – Riker and minimal winning coalitions was a huge literature • Economics – Shubik used a cooperative solution concept, the core, to demonstrate that the general equilibrium of Arrow-Debreu was also the core of the game in economics terms (see below) • The first formal type publication in Political Science was the Shapley- Shubik “value” applied to such as Congress and the UN N.B.: Shapely and Shubik were graduate students of Morganstern and of Nash

28. Payoff Configurations, Imputations, and Characteristic Functions • Payoff configuration – vector of utilities for any given outcome. • Imputations – Payoff configurations that are • Individually rational, i.e., give every one their security level 9what they can get playing rationally and alone) • Pareto Optimal • Characteristic Function – a form that assigns a real-valued payoff to each coalition.

29. Coalitions and the Core, Value, Bargaining Set • Core – • One definition is that the core of the game is the set of imputations that give each coalition their security level • Core is general eq. in economics; • R. Wilson – Arrow’s theorem is the “observations that the core of a voting game is generally empty” • Is the eq. concept in use in most spatial modeling (of the Davis-Hinich-Ordeshook-McKelvey sort)

30. Shapley Value • Shapley-Shubik Power Index – Measures “value” based on the incidence with which a member or coalition is “pivotal” that is, by its inclusion, changes a coalition from losing to winning. • Shapley Value is the more general version looking at the marginal increase in payoff to a coalition.

31. Bargaining Set and Competitive Solution • A large class of games that provides a solution to the game by which is meant a coalition structure and a payoff configuration

32. Win Set • Perhaps the most important of the cooperative style solution concepts still in use (social choice theory, in general, has been approached as cooperative game setting, with the win set one concept used) • Win set is defined (for a majority rule game) as W(x) consists of the set of points that defeat x by majority rule. If there exists a point, say, y for which the win set is empty, it is a preference induced equilibrium, according to Shepsle, and is a point in the core.

33. Non-Cooperative Game Theory The Dominant Form in Recent Decades

34. NC Game Theory Fundamentals • Player goals are represented by utility functions with utility defined over outcomes. • Actions and Strategies • A strategy is a plan of action. • In games that can be modeled as if they are simultaneous, actions and strategies are equivalent. • In other games, strategies and actions are quite different with strategies being the primary choice of interest. • The combination of actions by all players determines a payoff for each player.

35. Normal Form Games

36. Study Loaf Study 100,100 50,0 Loaf 0, 50 -10, -10 A normal form game • By convention, the payoff to the so-called row player is the first payoff given, followed by the payoff to the column player. Graduate School Student B Graduate School Student A

37. Practical Description • The normal form representation of a game specifies: • The players in the game. • The strategies available to each player. • The payoff received by each player for each combination of strategies that could be chosen by the players. • Actions are modeled as if they are chosen simultaneously. • The players need not really choose simultaneously, it is sufficient that they act without knowing each others’ choices.

38. Components of a Normal Form Game • Players A small number. • Actions Define columns and rows. • Strategies. Define columns and rows. • Information. Complete. • Outcomes. Represented by vectors in cells. • Payoffs. Elements of the vectors. • Equilibrium concept. Nash.

39. Technical Definition • 1 to n: players in an n-player game. • Si: player i’s strategy set. • Si: an arbitrary element of Si. • ui(si): player i’s payoff function. • Definition: The normal-form representation of an n-player game specifies the players’ strategy spaces S1,…,Sn and their payoff functions u1,…,un. • We denote the game by G={S1,…Sn;u1,…un}.

40. Nash Equilibrium • For an equilibrium prediction to be correct, it is necessary that each player be willing to choose the strategy described in the equilibrium. • Equilibrium represents the outcome of mutual and joint adaptation to shared circumstances. • If the theory offers strategies that are not a Nash equilibrium, then at least one player will have an incentive to deviate from the theory’s prediction, so the theory will be falsified by the actual play of the game.

41. Technical Definition • In the n-player normal-form game G={S1,…Sn; u1,…un}, the strategies (s1*,…sn*) are a Nash equilibrium if, for each player i, • s*i is (at least tied for) player i’s best response • to the strategies specified for the n-1 other players, (s*1,…s*i-1, s*i+1,…s*n): ui(s*1,…s*i-1,s*i, s*i+1,…s*n)≥ ui(s*1,…s*i-1, si, s*i+1,…s*n) • for every feasible strategy si in Si; • that is, s*i solves max siSi ui(s*1,…s*i-1, si, s*i+1,…s*n). • If the situation is modeled accurately, NE represent social outcomes that are self-enforcing. • Any outcome that is not a NE can be accomplished only by application of an external mechanism.

42. Left Middle Right Up 1,0 1,2 0,1 Down 0,3 0,1 2,0 Left Middle Right Top 0,4 4,0 5,3 Middle 4,0 0,4 5,3 Bottom 3,5 3,5 6,6 Elimination of dominated strategies Figure 1.1.1. Iterated domination produces a solution. Figure 1.1.4. Iterated elimination produces no solution.

43. Requirements for Iterated Domination • To apply the process for an arbitrary number of steps, we must assume that it is common knowledge that the players are rational. • We need to assume not only that all the players are rational, but also that all the players know that all the players are rational, and that all the players know that all the players know that all the players are rational, and so on, ad infinitum. • In the many cases where there is no or few strictly dominated strategies, the process produces very imprecise predictions.

44. Left Right Top 8, 10 -100, 9 Bottom 7, 6 6, 5 Left Middle Right Top 4, 3 5, 1 6, 2 Middle 2, 1 8, 4 3, 6 Bottom 3, 0 9, 6 2, 8 Example 1: A game with a dominated strategy. Example 2: A more complicated game: with dominated strategies.

45. NE: Fun facts • If iterated elimination of dominated strategies eliminates all but one strategy for each player, then these strategies are the unique NE. • There can be strategies that survive iterated elimination of strictly dominated strategies but are not part of any Nash equilibrium. • A game can have multiple Nash equilibria. The precision of its predictive power at such moments lessens. • Existence versus uniqueness

46. Left Right Top 4, -4 1, -1 Bottom 2, -2 3, -3 Solving for MS-NE • Row chooses “top” with probability p and bottom with probability 1-p. • Column chooses “left” with probability q and “right” with probability 1-q. • Players choose strategies to make the other indifferent. • 4q+1(1-q)=2q+3(1-q) • -4p-2(1-p)=-1p-3(1-p) • The MS-NE is: p=.25, q=.5. • The expected value of either Row strategy is 2.5 and of either Column strategy is –2.5

47. Mixed strategy NE • A mixed strategy Nash Equilibrium does not rely on an player flipping coins, rolling, dice or otherwise choosing a strategy at random. • Rather, we interpret player j’s mixed strategy as a statement of player i’s uncertainty about player j’s choice of a pure strategy. • Individual components of mixed strategy equilibriums are chosen to make the other player indifferent between all of their mixed strategies. • To do otherwise is to give others the ability to benefit at your expense. Information provided to another player that makes them better off makes you worse off.

48. The last word. • Theorem (Nash 1950): In the n-player normal-formgameG={S1,…Sn; u1,…un}, if n is finite and Si is finite for every i then there exists at least one Nash Equilibrium, possibly involving mixed strategies.

49. Extensive Form Games

50. Extensive Form Games • Allows dynamic games – player moves can be treated as sequential as well as simultaneous. • Complete information – games in which all aspects of the structure of the game –including player payoff functions -- is common knowledge. • Perfect information – at each move in the game the player with the move knows the full history of the play of the game thus far. • Imperfect information – at some move the player with the move does not know the history of the game.