Game Theory and The Constitution

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Game Theory and the Creation of the U.S. Constitution
People
Self interest
cooperation
Self Interest
1, 1
3, 0
Cooperation
3, 0
2, 2
People
Above is the classic Prisoner’s Dilemma. In order to ensure cooperation by both parties, it must be
enforced. This can be done with trust (morality), Government (authority), or Nash Equilibriums
When authority is chosen to enforce, this puts us in another Prisoner’s Dilemma with the authority
being a player. As with other players, authority (government) may also attempt to pursue their own self
interest.
Government
Self Interest
Self Interest
Cooperation
1, 1
3, 0
Revolution
Anarchy, Chaos, disorder
0, 3
2, 2
Dictatorship
Constitutions, equilibrium, laws, courts
People (states)
Cooperation
The Founding Fathers understood this and, as a result, created the Articles of Government which
allocated most power to the States, and left a limited amount for the Central Government.
Under the Articles of Confederation, the states had the ability to pursue their self interest without
restrictions. They were under no obligation to cooperate with either the central government or the
other states. Creating a national government with no executive or judicial authority left the central
government powerless in enforcing cooperation amongst the states. A prisoner’s dilemma between the
states emerged with only trust acting as the enforcement mechanism. Under this format, the Articles
were destined to fail. It was a contract- not a coordinated Constitution. Meaning, it only worked when
it was in the states best interest to work together. Without an outside unifying need (Revolution) states
were free to follow or break the contract.
Interestingly enough, the Articles worked better during the Revolution as compared to times of peace.
This was because it was in the individual state’s best interest to cooperate against the British. The end
of the war also resulted in an end for the need to cooperate. After the war, the Articles became a true
Prisoner’s Dilemma.
Article of Confederation during War
Articles during peace
(C/C= Pareto Dominate)
(D/D equilibrium results from iterating game)
States
D
States
D
C
0, 0
0,2
States
D
C
D
1, 1
3, 0
C
0, 3
2, 2
States
C
2,0
3,3
The diagram above demonstrates an interesting point about Constitutions and equilibriums in general.
Successful Constitutions result when various groups agree not to defect because of their self interest
and not because of law, morals, and trust.
Successful Constitutions (we have the longest lasting to date) are not contracts, but are instead
coordinated documents. Our Constitution is a series of coordinated equilibriums between the states
and the federal government, the people and the government, and the different branches. This
coordination creates many Nash Equilibriums.
Nash Equilibrium- In game theory, Nash equilibrium (named after John Forbes Nash, who proposed it) is
a solution concept of a game involving two or more players, in which each player is assumed to know
the equilibrium strategies of the other players, and no player has anything to gain by changing only his
own strategy unilaterally. If each player has chosen a strategy, and no player can benefit by changing his
or her strategy while the other players keep theirs unchanged, then the current set of strategy choices
and the corresponding payoffs constitute Nash equilibrium.
In other words, a Nash Equilibrium occurs when each player plays their best response taking into
account the other player’s decision, and vice versa.
The founding fathers knew from experience that the best way to enforce the cooperation is to create a
self-enforcing document with multiple Nash Equilibriums. This document is more of a coordination
problem than it is a contract.
= Nash Equilibrium
Below are examples of Coordination problems.
Choosing sides- A/B- Dominate
A
A
10,10
B
0, 0
Pure Coordination- A is Pareto Dominate
B
A
B
0,0
A
10, 10
0,0
10, 10
B
0,0
5,5
Stag Hunt
Battle of the Sexes (conflicting interest coordination)
A
B
A
B
A
10,5
0, 0
A
10, 10
0,7
B
0, 0
5, 10
B
7, 0
7, 7
The Principles of the Constitution (Federalism, limited government, popular sovereignty, separation of
powers, and checks and balances) are intended to put all “players” of the game in Nash Equilibriums.
Players in the game- (the people, states, national government, and the different branches): Below is a
list of examples:
Voting
Check and balances
War Powers
Taxing
Representation
Amendment processes
State vs Individual
Structure of Congress
Bill of Rights
Necessary and Proper Clause vs the 10th Amendment and article I section 9
The question of unwritten power
Term limits
Re-election
Electoral College
Game Theory and the Constitutional Convention and Ratification
Game theory with the voting procedures of the delegates
States are in an Equilibrium situation
Rhode Island refused to send Delegates
Northern States
Southern States
Big States
Small States
Massachusetts
New Jersey
New York
New Hampshire
Pennsylvania
Connecticut
Virginia
Georgia
North Carolina
Delaware
South Carolina
Maryland
Each Delegate voted to best promote their self interest. But if the delegations could not agree on a
decision, then they will be forced to abstain. This would alter the equilibrium situations and shift the
balance of power away from the delegate’s state and region.
Each state needed to put themselves in the other’s shoes, eliminate weakly dominated strategies and
decide on the best response from the remaining options.
The situation above forces each delegation into a prisoner’s dilemma when delegates cannot agree with
each other.
When the self interest of each delegate from the same states comes in conflict:
Delegate 2
Self Interest
Self Interest
Cooperation
1, 1
3, 0
0, 3
2, 2
Delegate 1
Cooperation
If both delegates seek their own self interest, then they will abstain from the voting and neither idea will
be presented, jeopardizing the overall interest of the state. As we have learned, this is possible if the
other party tries to take advantage of the situation. The compromise of Cooperation/Cooperation is
always threatened by the aggressive solution of Self Interest/Self Interest (Pareto dominated)
In this situation, Self Interest represents a Nash’s Equilibrium. A Nash Equilibrium describes a situation
when both parties choose the best strategy, given the strategy chosen by the other participant.
However, both delegates realize that they will have many topics to vote on, and therefore a repeating
game is the result. A delegate will hurt future cooperation if they defeat from the present decision. As
a result, it is possible, and in this situation- advisable for both parties to cooperate. Otherwise, they may
never have a say in the convention. They must “coordinate” their vote.
Delegate 1
C
D
D
D
D
D
Delegate 2
D
D
D
D
D
D
This leads us into a coordination games, Coordination problems are a class of games with multiple Nash
Equilibriums in which players choose the same or corresponding strategies. They choose corresponding
strategies because they can only receive mutual gains by making mutually consistent decisions.
When both parties agree this becomes a Pure Coordination game
(In this example let’s assume both delegates want “yes”)
Delegate 2
Yes
Yes
No
10, 10
0, 0
Pareto Dominate
Nash equilibrium
Delegate 1
No
0, 0
5, 5
Nash Equilibrium
When both parties cannot agree but they realize that it is better to compromise (even if it means deals
for future voting or amendments) than it is to abstain.
This becomes a Conflicting Interest Coordination (Battle of the Sexes)
In this example, Delegate 2 prefers “yes,” Delegate 1 prefers “no” neither wants to abstain. Both
delegates agree to vote yes through compromise (you figure it out) this is what the matrix looks like.
Delegate 2
Yes
Yes
No
1, 2
0, 0
0, 0
2, 1
Delegate 1
No
Every thing beyond this, must be left to the prudence and firmness of the people; who, as they will hold
the scales in their own hands, it is to be hoped, will always take care to preserve the constitutional
equilibrium between the General and the State Governments
Look for equilibrium in the document itself.
Make a list of Equilibriums in the Constitution.
Ratification
Coordination problems necessitate communication. This is why the Federalist papers become very
important. To be ratified, the Constitution needed to be ratified by 9/13 states. Communication was a
must.
But what happens in the absence of communication (the late 1780’s wasn’t a time filled with
newspapers or CNN)? Or maybe even worse, how could this be ratified in the mist of bad
communication or with information that said ratification was a bad idea?
The answer is Focal Points.
In game theory, a focal point (also called Schelling point) is a solution that people will tend to use in the
absence of communication, because it seems natural, special or relevant to them. The idea originated in
the 1960’s by Thomas Schelling.
The Constitution is filled with many focal points and where states “met” at these points of equilibrium.
There are many focal points, start with looking at the principles of Popular Sovereignty, limited
government, federalism, separation of powers, and checks and balances.
Additionally, there was a missing part of the Constitution where almost all states and people naturally
agreed with each other- another focal point. This was the insistence of a Bill of Rights.
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