Extensive form games and an application: “Economic Origins of Dictatorship and

Democracy”

Presentation for Political game theory reading group

Carl Henrik Knutsen, June 2008

Extensive form• Way to model games. More appropriate than normal form in games

with dynamic elements• Terminology:

– Nodes (e.g. initial, terminal), point in game where a specific actor can act– Branches, illustrates possible actions for an actor from a given node

• Consists of description of – Set of Agents – Set of Histories (up until a node)– A mapping of which actors that can make choice at different nodes– All possible actions at a given node– Pay-offs– Information set: IS specifies players’ information at decision nodes in game. A

IS is a set of nodes between which a player cannot distinguish when making a decision

The game treeA

No Threat Threat

(1,1) B

Give in Fight

(0, 2) (1-c, 1-c)

Backward induction

• Sequential rationality: optimizing at all info.sets where a player can move

• Procedure: Start with last player (B) to act. Find optimal action. Player that acts before (A) knows that B is rational. A then chooses actions optimally, given his knowledge of Bs optimal action at the later stage.

• In general: Continue this process up the game tree until we reach initial node.

Subgame

• Given an extensive form game, a node x in the tree is said to initiate a subgame if neither x nor any of its successors are in an information set that contains nodes that are not successors of x.

• All players must “know where they are” in an initial node of a subgame. Cannot split information sets

• Subgames as self-contained extensive forms.• Some examples

Subgame Perfect Nash Equilibrium

• Recognize that we have two NE in the game:(No threat, Fight) and (Threat, Give in) • However, one of the equilibria is implausible if we assume

sequential rationality• We need a solution concept that is more appropriate, (can take

away implausible equilibria) in dynamic games: Subgame Perfect Nash Equilibrium

• Definition: A strategy profile is a SPNE if it specifies a Nash Equilibrium in every subgame of the original game.

• Use definition to show that (No threat, Fight) is not a SPNE• Common application: Credible and non-credible threats and

promises

Imperfect information

• Imperfect information if at least one of the info.sets in the game is not a single node

• That is: At least one of the players at at least one point in the game does not know the full history of the game/where he is

• Have to modify our strategy of backward induction: Find subgames and use SPNE as solution concept.

Acemoglu and Robinson (2006)

• “A framework for analyzing the creation and consolidation of democracy”

• Three fundamental building blocks (page xii)– “Economics based approach”. Economic incentives

drive behavior and strategic actors– Importance of conflict and opposing interests

between actors/groups– Political institutions are important in solving

commitment problems by affecting future power distribution

Framework

• Two groups: elites and citizens (alt. rich and poor)

• Regime-types: right-wing dictatorship and democracy (+ left-wing dictatorship)

• In democracy: Median voter determines policies

• In right-wing dictatorship: Elites determine policy

• Question: Why do elites democratize?

Some notes

• Commitment problem is crucial to A&R, there exist credible and non-credible commitments. Credible commitments need political institutions to back them up. Follows in the tradition of for example North&Weingast

• A&R also looks at coups (democracy-reversals), but these will not be treated here. Same framework and logic.

• In book: more complex mathematics. Optimizing over infinite time horizon, given different states, Bellman-equations, Markov-chains etc

• Model presented in M&M is a simple version mathematically, and the framework is as parsimonious as it gets.

The model

• Two agents, rich and poor. Population normalized to 1. Share of poor = λ>½

• Average income is y = λyp+(1-λ)yr, where yp<y<yr

• θ is share of income going to the poor yp=θy/λ and yr=((1-θ)y)/(1-λ)

• Policy instrument is linear tax rate, τ, and lump sum redistribution. (As simple as it gets)

• Transaction/distortionary cost of taxation: C(τ)y. C’>0 and C’’>0. Set C(τ) = ½τ2 total tax revenue, T=(τ- ½τ2)y

The model

• After-tax income for type i є {r,p}:• Vi(τ)= (1-τ)yi+(τ- ½τ2)y• Maximize income with respect to τVp’(τ)=y-yp-τy=0• Since yp=θy/λ, τp*=(λ-θ)/λ (between 0 and 1)• Similar calculation for rich implies negative

tax rate optimal tax rate is set to 0

Actions• Rich decides first whether to democratize or not. Then poor

observes choice and decides whether to conduct revolution or not. Revolution is costly, but means that the poor can implement their favored regime, a Marxist dictatorship where the rich have income=0

• In democracy: Poor will set optimal tax rate as deduced above, because they include median voter. In right-wing dictatorship, rich set favored tax-rate, which is zero

• Cost of revolution is contingent on political shock, s: μs S can be h or l we have μh>μl. Interpretation: strength of regime, international context, ease of coordinating revolution (the importance of solving collective action problems)..

• 1-μs modelled as destroyed income for the total economy.

Solving the game; backward induction

• Rich consider “the revolution constraint”: Make sure that poor do not prefer revolution. – If revolution is very costly (low shock), the poor will not

revolt in any case Rich will recognize this and decide not too democratize.

– If revolution is less costly, poor will revolt if right-wing authoritarian. Rich recognize this and democratize. Poor will now not have the incentive to revolt, since they earn relatively more under democracy.

– If revolution costs extremely little, poor will revolt in any case and split the income between themselves and leave the rich with nothing

More precisely..

• Revolt against right-wing if Vp(R,μs)>Vp(0)• If however Vp(R,μs)< Vp(τp*), poor can be

appeased by democratization, and rich will democratize since Vr(τp*)>0

• The importance of the nature of the shock.. Vp(R,μs)>Vp(0)μs>θ..The shock matters if μh>θ>μl

• Revolt against democracy if: Vp(R,μs)>Vp(τp*) μs>θ + (λ-θ)2/2λ

• λ>θ and the second term is therefore positive, which means that a higher shock (smaller destruction) is necessary to revolt against a democracy than against a right wing dictatorship

The game tree

Rich

D N Poor Rich

τ τ

Poor Poor

R NR R’ NR’

0, Vp(R,μs) Vr(τp*), Vp(τp*) 0, Vp(R,μs) Vr(N,0), Vp(N,0)

The commitment problem• Why can’t the rich elite just increase taxes and redistribution without

democracy? A&R: more than one period. Rich cannot commit to keep the high tax rate once the revolutionary threat is gone (low shock in t+n, collective action problems), and the poor know this.

• Difficult to model in the game presented here. M&M’s solution is to say that the rich have a specified probability of keeping their promise of redistribution under dictatorship. – P increases in revolutionary threat. – P increases with inequality (lower θ). Logic: High inequality means that the poor

have little to lose by revolting. The rich recognize this and redistributes (with a certain p) so that the poor are indifferent between revolting and accepting the existing regime. A perhaps counterintuitive implication is therefore that higher inequality decreases the probability of democratization, since the permanent revolutionary threat now implies that the elite can credibly commit to redistributing. IT IS THE SEQENTIAL NATURE OF THE GAME THAT DRIVES THIS RESULT. Notice that the rich lose more by democratizing in an inegalitarian society because the median voter is poorer

• The algebra can be found in M&M pages 196-7.