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Introduction to Game Theory: Nash Equilibrium and Strategic Decision Making

Learn about Nash Equilibrium, strategic choices, and Game Theory in various scenarios to understand decision-making strategies. Delve into simulations and predictions to grasp the concept effectively.

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Introduction to Game Theory: Nash Equilibrium and Strategic Decision Making

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  1. Social Choice Session 7 Carmen Pasca and John Hey

  2. 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).

  3. 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.

  4. 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.

  5. 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).

  6. 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).

  7. 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).

  8. 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.

  9. 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.

  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.

  11. 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)

  12. 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.

  13. 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.

  14. 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.

  15. 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).

  16. 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!

  17. 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?

  18. 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).

  19. 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?

  20. 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.

  21. 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?

  22. 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???

  23. 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?

  24. 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!

  25. 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)

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