Social Choice Session 7

Carmen Pasca and John Hey

Strategic Decision Making: Game Theory

• Later on in the course (sessions 15 and 16) we will turn to Social Contract Theory.

• Some of that material will refer to the book by Ken Binmore Game Theory and the Social Contract (volumes 1 and 2) in which he draws heavily on Game Theory.

• To prepare ourselves for that, and for the material on Public Goods in sessions 8 and 9 (and some of session 10), we will in this session study the rudiments of Game Theory.

• We will study simple static games, sequential games, dynamic (repeated) games, and possibly evolutionary games.

• The key concept is that of a Nash Equilibrium in such games (not that we necessarily agree with it).

We start simple

• We begin by considering two person simultaneous-play games where each player has just two choices.

• We present the game with a payoff matrix. For example:

• Note that this is colour-coded.

Player 2

Left Right

Player 1

Up 16,14 10,12

Down 4,6 2,0

What does this mean?

• Let us define the rules and take our time. • Players 1 and 2 choose independently and simultaneously.• They get the payoffs colour-coded in the table.

• For example, if Player 1 chooses Up and Player 2 chooses Right then Player 1 gets payoff 10 and Player 2 gets payoff 12.

Player 2

Left Right

Player 1

Up 16,14 10,12

Down 4,6 2,0

Prediction in this game

• Here it is simple:

• Whatever player 2 chooses, it is best for Player 1 to choose Up.• Whatever player 1 chooses, it is best for Player 2 to choose Left.• Both players have a dominating choice. Prediction: (Up,Left).

Player 2

Left Right

Player 1

Up 16,14 10,12

Down 4,6 2,0

Here it is less simple

• Whatever player 2 chooses, it is best for Player 1 to choose Up.• If Player 1 chooses Up (Down), it is best for Player 2 to choose Left (Right).• What should Player 2 do? • He/she can work out that Player 1 is going to choose Up so it is best for

Player 2 to play Left.• Prediction: Up, Left. This is a Nash Equilibrium (defined later).

Player 2

Left Right

Player 1

Up 16,14 10,12

Down 4,6 2,10

Even less simple

• If Player 1 chooses Up (Down), it is best for Player 2 to choose Left (Right).• If Player 2 chooses Left (Right), it is best for Player 1 to choose Up (Down).• Here Nash steps in. He defines a Nash Equilibrium as when each

player is choosing his or her best given the choice of the other.• Here we have two Nash Equilibria (Up,Left) and (Down,Right).• In the previous two games just one both at (Up,Left).

Player 2

Left Right

Player 1

Up 16,14 2,12

Down 4,6 8,10

Nash Equilibrium

• A general definition:• A Nash Equilibrium in a two-person game is when each player is doing what is

best for him or her given what the other player is doing.• In this latter game there are two Nash Equilibrium.• These are both NE in Pure Strategies.• We can also have NE in Mixed Strategies – here players play probabilistically.

There is an example in the next slide.• Note that in this example there are no Nash Equilibria in pure strategies.

•

Mixed Strategy Nash Equilibrium

• Note that there are no Nash Equilibria in pure strategies.• However there is a Nash Equilibrium in mixed strategies

with Player 1 playing Up and Down with equal probabilities and Player 2 playing Left and Right with equal probabilities.

Player 2

Left Right

Player 1

Up 10,-10 -10,10

Down -10,10 10,-10

Asymmetric Mixed Strategy Nash Equilibrium

• Again there is no Nash Equilibrium in pure strategies.• Suppose Player 1 plays Up with probability p and Player 2

plays Left with probability q. These must satisfy:• 1q + 0(1-q) = 0q+ 3(1-q) and 2p + 5(1-p) = 4p + 2(1-p). (Why?)

• Hence p = ⅗ and q = ¾.

Player 2

Left Right

Player 1

Up 1,2 0,4

Down 0,5 3,2

The Prisoner’s Dilemma

• This is a very famous example. It is very similar, as we will see, to the Public Goods game which we shall study in detail later.

• We start with general values where a>c d>b and b>a.

• Is there a unique NE in pure strategies? YES (Up,Left)

Player 2

Left Right

Player 1

Up a,a d,c

Down c,d b,b

Now let us take some particular numbers

• Take a=10, c=0,b=20,d=30

• (Up,Left) is the unique Nash Equilibrium.• Note that both players have a dominant strategy.• Note that (Down,Right) Pareto Dominates this. (Better for both.)

• But it is not a Nash Equilibrium.

Player 2

Left Right

Player 1

Up 10,10 30,0

Down 0,30 20,20

Now let us take some other numbers

• Take a=10, c=0,b=1000,d=1010

• Again (Up,Left) is the unique Nash Equilibrium.• Again that both players have a dominant strategy.• Again (Down,Right) pareto dominates this.• Again it is not a Nash Equilibrium.

Player 2

Left Right

Player 1

Up 10,10 1010,0

Down 0,1010 1000,1000

Note the oddity

• The unique Nash Equilibrium is Pareto Dominated by (Down,Right) ….

• …the latter is simply better for both players…• …but both have a direct incentive to play in a way that they

end up at the Nash Equilibrium.• What about pre-play communication?• Both have an incentive to cheat on any pre-play agreement

unless it is binding in some way.• What about repetition?• Let us suppose we play the game n times.• To find the solution we need to backward induct.

Playing the game n times (n fixed and known)

• Players use Backward Induction• In the nth play we get (Up,Left) as the NE.• because of this in the (n-1)th play we get (Up,Left).• …• because of this in the 1st play we get (Up,Left).

Player 2

Left Right

Player 1

Up 10,10 1010,0

Down 0,1010 1000,1000

Other possibilities

• If we play the game an infinite or a random number of times we might get (Down,Right).

• Also we might get what is called a Tit for Tat strategy.• In this Player 1 plays Down until Player 2 plays Left,

and plays Up until Player 2 plays Right…• …and Player 2 plays Right until Player 1 plays Up, and

plays Left until Player 1 plays Down.• What happens with this Tit for Tat strategy is that we

end up at (Down,Right) all the time.• Jolly Good!

Change the rules from simultaneous to sequential

• Player 1 goes first and then Player 2: what happens?• We end up still at (Up,Left). Why?• Player 2 goes first and then Player 1: what happens?• We end up still at (Up,Left). Why?• Is this always true in whatever the game?

Player 2

Left Right

Player 1

Up 10,10 1010,0

Down 0,1010 1000,1000

Other Games

• We have studied the solution concepts and some interesting games.

• Other games are of interest for economics.• These include:• Coordination Games (for example, Battle of the

Sexes).• Games of Competition.• Games of Coexistence.• Bargaining Games (we will meet these again later).

Battle of the Sexes (Coordination Game)

• Two NE in Pure Strategies: (Up,Left) and (Down,Right).

• Mixed Strategy NE with p=2/3 and q=2/9.• If you had to predict behaviour in this game, what

would you predict?

Player 2

Left Right

Player 1

Up 8,4 2,1

Down 1,2 4,8

Competition Game

• If x<0 then NE is (Up,Left)• If 0<x<1 then NE is (Down,Left)• If x>1 no NE in pure strategies – there is a mixed NE

with p and q depending on x.

Player 2

Left Right

Player 1

Up 0,0 2,-2

Down x,-x 1,-1

The Hawk-Dove Game (Coexistence Game)

• Players are Bears. • Up and Left is Hawk and Down and Right is Dove.• Pure NE are (Left,Down) and (Up,Right). [(Down,Right) is not.]• There is a mixed NE with p=4/9 and q=4/9.• Prediction?

Player 2

Left Right

Player 1

Up -5,-5 8,0

Down 0,8 4,4

The Public Good Game

• We will come across this later.• There are N people in society, each with an income m.• They each independently and simultaneously and without

communication choose an amount x to donate to the ‘public good’.

• The total amount donated is multiplied by some number k and distributed equally to everyone.

• So individual n gets (m-xn) – what is left over of his income – plus k(x1+…+x1)/N.

• What do people contribute?• Game theory/Nash Equilibrium???

Let us take a simple example

• Three people in society (N=3) each with income 100.• Suppose the multiplier, k, is 2.• Suppose 1 donates 0, 2 donates 50 and 3 donates 100.• Total public good is 150. Multiplied by 2 it becomes 300. • When distributed equally everyone gets 100.• Thus:• 1 ends up with 100+100 = 200• 2 ends up with 50+100 = 150• 3 ends up with 0+100 = 100.• What do you think is the Nash Equilibrium?• What is the Social Optimum?

An amusing evolutionary game

• Two Players repeatedly playing the Coordination Game shown earlier.

• Eventually they learn to coordinate on one of the two NE.

• A variant: before each repetition either a Yellow light flashes or a Green light flashes.

• Note there is no information in the light.• Experimental evidence shows that behaviour gets to

a NE even quicker!• Weird!

Conclusion

• Game theory is based on Nash Equilibrium…• …everyone doing what is best for them given what

everyone else is doing.• We have seen that it leads to what we would call

‘antisocial’ outcomes.• Conclusion?• If Game Theory is correct, the state needs to intervene

(in, for example, the provision of Public Goods).• But is it correct?• Everywhere? (Sweden/Italy)