Basic elements of game theory

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Basic elements of game theory

Game Theory - Motivation

• Game Theory is a tool to understand situations in which two or more decision-makers interact.

• Outcomes of economic decisions frequently depend on others’ actions

– effect of price policy depends on competitors

– outcome of wage negotiations depends on choices of both sides

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• Decision makers should thus take expectations of others’ decisions into account

• Such situations are plausibly modeled as a “game”, a model of interactions where the outcome depends on others’ as well as one’s own actions • this definition and the scope of game theory is

much broader than the everyday definition of a game

– e.g., game theory is not only concerned with “winning” a competitive game

Definition of a game

A game is a formal representation in which a number of agents interact in a setting of strategic interdependence. By that, we mean that each individual’s welfare depends not only on her own actions but also on the actions of the other individuals.

Moreover the actions that are best for her to take may depend on what she expect the other players to do

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To describe such situations we need four things:

1) The players: who is involved?

2) The rules: Who moves when? What do they know

when they move? What they can do?

3) The outcomes: for each possible set of actions by

the players, what is the outcome of the game?

4) The payoffs: what are the players’ preferences over

the possible outcomes?

Example: Matching Pennies v. 1 1) The players: John and Paul

2) The rules: Each player simultaneously puts a penny

down either head up or tail up

3) The outcomes: if the two pennies match John pays

1 euro to Paul, otherwise Paul pays 1 euro to john

4) The payoffs: Paul and John prefer receive 1 euro

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2) The rules: John puts a penny down either head up

or tail up. Paul observes the decision of John and puts a penny down either head up or tail up.

instead to pay

2) The rules: John puts a penny down either head up

or tail up. After 1 minute the decision of John and without the possibility to know the decision of John, Paul puts a penny down either head up or tail up.

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Extensive – Form Representation

An extensive form representation of a game specifies: • Players

• When each player has to move

• The actions a player can use at each of his opportunities to move

• What a player knows at each of his opportunities to move

• Payoffs received by each player for each possible outcome

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Game Trees

• An extensive form game can be represented in a

game tree

• This shows

– who moves when (at the nodes) – available actions (the branches) – available information

– the payoffs over all possible outcomes (at the

terminal nodes)

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Example 1

Player 1 L R Player 2 Player 2 l r l r U1(L,l) U2(L,l) U1(L,r) U2(L,r) U1(R,l) U2(R,l) U1(R,r) U2(R,r)

The preferences can be represented by a payoff function over the outcomes

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Example 2

Player 1 L R Player 2 Player 2 l r l r U1(L,l) U2(L,l) U1(L,r) U2(L,r) U1(R,l) U2(R,l) U1(R,r) U2(R,r) 12 This dashed line means that player 2 does not know the action played by player 1

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Information set

• It is a collection of decision nodes where:

– The player has to move at every node in the information set

– When a player has to move, he cannot distinguish the nodes belonging to the same information set – Example 1: player 1 has one info set, player 2 has

two info sets

– Example 2: player 1 has one info set, player 2 has one info set

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Note: What can an info set NOT look like

Player 1 Left Right Player 2 l r l _m r Player 1 14

The two nodes in the information set have different number of available actions, then player 2 can distinguish the node

This could be true only assuming that player 1 does not remember his move in the first node

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Strategies

• A strategy is a complete description of a player’s actions in all the information sets where he has to move, e.g.

– for player 2 to choose r after L and l after R, i.e. (r, l) . – Player 2 has 4 strategies: {(l,l),(l,r),(r,l),(r,r)}

Player 1 L R Player 2 Player 2 l r l r U1(L,l) U2(L,l) U1(L,r) U2(L,r) U1(R,l) U2(R,l) U1(R,r) U2(R,r) 15

A subgame starts at an information set with a single node n - it contains all decision and terminal nodes following n - an information set cannot belong to two different

subgames

Note: someone considers the whole game a subgame,

others do not consider the whole game a subgame. In the following we use the first approach

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• perfect information, i.e. when choosing an action a player knows the actions chosen by players moving before her

i.e. all previous moves are observed before the next move is chosen

• Imperfect information when at least one player does not know the history by the time he chooses.

At least one player does not know all the actions chosen by players moving before him

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Dynamic games of perfect and imperfect

information

In other words, when a game is of imperfect info, there exists at least an information set with more than one decision node Player 1 Left Right Player 2 l r l r U1(L,l) U2(L,l) U1(L,r) U2(L,r) U1(R,l) U2(R,l) U1(R,r) U2(R,r) 18 Player 2 knows that he is in the information set, but not in which specific node

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Example 1: Mini Ultimatum Game

• Proposer (Player 1) can suggest one of two splits of £10: (5,5) and (9,1).

• Responder (Player 2) can decide whether to accept or reject (9,1), but has to accept (5,5). Reject leads to 0 for both Player 1 (9,1) a r 9 1 0 0 (5,5) 5 5 Player 2 19 Perfect information

Player 1 has one information set Player 2 has one information set two subgames

Example 2

Imperfect information

Player 1 has one information set Player 2 has one information set One subgame Player 1 Left Right Player 2 l r l r U1(L,l) U2(L,l) U1(L,r) U2(L,r) U1(R,l) U2(R,l) U1(R,r) U2(R,r) 20

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Example 3

Player 1 has one information set Player 2 has two information sets Two subgames Player 1 Left Centre Player 2 l r l r U1(L,l) U2(L,l) U1(L,r) U2(L,r) U1(C,l) U2(C,l) U1(C,r) U2(C,r) Right Player 2 U1(R,l) U2(R,l) U1(R,r) U2(R,r) 21 l r

Example 4

Challenger: one information set Incumbent: one information set One subgame Challenger Ready Unready Incumbent Fight 3 3 1 1 4 3 0 2 Out 2 4

Acquiesce Acquiesce Fight

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Challenger Ready Unready Incumbent Fight 3 3 1 1 4 3 0 2 Out 2 4

23 Challenger

In

Example 5

Challenger: two information sets Incumbent: one information set Two subgames

Normal form representation

A normal form specifies:

1. the agents in the game,

2. for each agent, the set of available actions (or strategies):

S_i denotes the set of strategies available to player i and s_i is an element of S_i

3. The payoff received by each player for each combination of strategies:

– u_i(s₁ … s_i … s_n) denotes the payoff function of player i where (s₁ … s_i … s_n) are the actions chosen by the players

Then a game can be represented as:

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Representation of a sequential game using

the normal form

Case of two players: 1 and 2

Label the rows of the normal form with the player 1’s strategies

Label the columns of the normal form with the player 2’s strategies

Compute the payoffs to the players for each possible combination of strategies

Using he normal form representation is possible to find all Nash equilibrium

25 Challenger Ready Unready Incumbent Fight 3 3 1 1 4 3 0 2

Acquiesce Acquiesce Fight

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Player 1’strategies: {Ready, Unready}

Player 2’strategies: {(A, A), (A, F), (F, A), (F, F)}

Example 6

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Player 1’strategies: {Ready, Unready}

Player 2’strategies: {(A, A), (A, F), (F, A), (F, F)} Incumbent

(A, A) (A, F) (F, A) (F, F) Challenger Ready 3, 3 3, 3 1, 1 1, 1

Unready 4, 3 0, 2 4.3 0, 2 Three Nash equilibria

1. Ready, (A, F) 2. Unready, (A, A) 3. Unready, (F, A) Player 1 Left Centre Player 2 l r l r U1(L,l) U2(L,l) U1(L,r) U2(L,r) U1(C,l) U2(C,l) U1(C,r) U2(C,r) Right Player 2 U1(R,l) U2(R,l) U1(R,r) U2(R,r) 28 l r Player 2 l, l l, r r, l r, r Player 1 Left U1(L,l), U2(L,l) U1(L,l), U2(L,l) U1(L,r), U2(L,r) U1(L,r), U2(L,r) Centre U1(C,l), U2(C,l) U1(C,l), U2(C,l) U1(C,r), U2(C,r) U1(C,r), U2(C,r) Right U1(R,l), U2(R,l) U1(R,r), U2(R,r) U1(R,l), U2(R,l) U1(R,r), U2(R,r)

Example 7

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Example 8

Challenger Ready Unready Incumbent Fight 3 3 1 1 4 3 0 2 Out 2 4

29 Incumbent Acquiesce Fight Challenger Ready 3, 3 1,1 Unready 4, 3 0, 2 Out 2, 4 2, 4 Challenger Ready Unready Incumbent Fight 3 3 1 1 4 3 0 2 In 2 4

30 Challenger In

Example 9

Incumbent Acquiesce Fight Challenger In Ready 3, 3 1,1 In Unready 4, 3 0, 2 Out Ready 2, 4 2, 4 Out Unready 2, 4 2, 4 Out

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Simultaneous games

Definition of “Best Response”

The Best Response of a player is his preferred action given the strategies played by the other players.

• Consider the n-player normal form game 𝐺 = 𝑆₁, … . . 𝑆_𝑛; 𝑢₁ … 𝑢_𝑛 • The best response of player i to the strategies

𝑠_−𝑖 = (𝑠₁, … , 𝑠_𝑖−1, 𝑠_𝑖+1, 𝑠_𝑛) solves the following problem:

max

𝑠_𝑖∈𝑆_𝑖𝑢(𝑠1, … , 𝑠𝑖, … , 𝑠𝑛)

We denote the set of best responses to 𝑠−𝑖 as 𝐵𝑅(𝑠−𝑖)

Example 1

Player 2 L C R T 2,3 2,2 5,0 Player 1 Y 3,2 5,3 3,1 Z 4,3 1,1 2,2 B 1,2 0,1 4,4

The best response of player 1 to player 2 playing L is Z, i.e. 𝑍 = 𝐵𝑅(𝐿) The best response of player 1 to player 2 playing C is Y , i.e. 𝑌 = 𝐵𝑅(𝐶) The best response of player 2 to player 1 playing B is R , i.e. 𝑅 = 𝐵𝑅(𝐵)

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Suppose a game with two players, 1 and 2. payoff of player 1 is

(10 – 𝑎₁– 𝑎₂) 𝑎₁

where 𝑎1is the action of player 1 and 𝑎2is the action of

player 2

What is player 1’s best response if player 2 plays 𝑎₂ = 3 ? To find it we have to solve

𝑀𝑎𝑥 (10– 𝑎₁– 𝑎₂)𝑎₁ The FOC are: 10 – 2𝑎₁– 𝑎₂ = 0

Solving by a1 we get player 1’best response 𝑎₁= (10 − 𝑎₂)/2

Example 2

Nash Equilibrium

• It is a prediction about the strategy each player will choose

• This prediction is correct if each player’s predicted strategy is a best response to the predicted strategies of the other players.

• Such prediction is strategically stable or self enforcing: no player wants to change his/her predicted strategy • We call such a prediction a Nash Equilibrium.

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Consider the n-player normal form game

G = {S₁,….. S_n; u₁… u_n}

The strategy profile

𝑠₁∗, 𝑠₂∗, 𝑠₃∗, … , 𝑠_𝑛∗ is a Nash equilibrium if:

for every player i and every action s_iS_i:

𝑢_𝑖 𝑠₁∗, … , 𝑠_𝑖∗, … , 𝑠_𝑛∗ ≥ 𝑢_𝑖 𝑠₁∗, … , 𝑠_𝑖, … , 𝑠_𝑛∗ i.e.

in a Nash equilibrium, each player strategy is a best response to the other players' strategies

in a Nash equilibrium there are no players that want to deviate

Example 1: the Prisoner’s Dilemma

Player 2

C(ooperate) D(efect)

Player 1 C(ooperate) 2,2 0,3

D(efect) 3,0 1,1

• The unique Nash equilibrium is (D,D)

• For every other profile, at least one player wants to deviate

• For Player 1 defect is a best response to Player 2 playing defect. For Player 2 defect is a best response to Player 2 playing defect

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Example 2: the “Battle of the Sexes”

Player 2

Football Theatre

Player 1 Football 2,1 0,0

Theatre 0,0 1,2

• There are two Nash equilibria: (Football, Football) and (Theatre, Theatre)

• For Player 1 Football is a best response to Player 2 playing Football. For Player 2 Football is a best response to Player 2 playing Football.

• For Player 1 Theatre is a best response to Player 2 playing Theatre. For Player 2 Theatre is a best response to Player 2 playing Theatre.

Example 3: Matching Pennies

Player 2

Head Tail

Player 1 Head 1,-1 -1,1

Tail -1,1 1,-1

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Example 4: “Stag-Hunt”

Player 2

Stag Hare

Player 1 Stag 2,2 0,1

Hare 1,0 1,1

• There are two equilibria: • (Stag, Stag) and (Hare, Hare)

Example 5: Public good game

This is an example of a game with continuous strategies Two individuals are endowed with 10 pounds.

They can contribute to a public good by delivering any amount of money out of their endowment (𝑐1, 𝑐2)

For each individual the value of the public good is given by the sum of the contributions multiplied by 0.7

𝑣𝑝 = 0.7 𝑐1+ 𝑐2

Payoff of player 1 =

= Endowment – contribution + value of the public good 𝜋1= 10 − 𝑐1+ 𝑣𝑝 = 10 − 𝑐1+ 0.7 𝑐1+ 𝑐2

Payoff of player 2 =

= Endowment – contribution + value of the public good 𝜋₂ = 10 − 𝑐₂+ 𝑣_𝑝 = 10 − 𝑐₂+ 0.7 𝑐₁+ 𝑐₂

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The best response of player 1 is given by the solution of the problem: max 𝑐₁ 𝜋1= 10 − 𝑐1+ 0.7 𝑐1+ 𝑐2 FOCs are: −1 + 0.7 < 0 Then player 1’s best response is 𝑐1 = 0

The best response of player 2 is given by the solution of the problem: max 𝑐2 𝜋₂= 10 − 𝑐₂+ 0.7 𝑐₁+ 𝑐₂ FOCs are: −1 + 0.7 < 0 Then player 2’s best response is 𝑐2= 0

Unique Nash equilibrium: both individuals contribute by 0 But the efficient outcome is both contributing by 10

Definition:

Subgame perfect Nash equilibrium

A Nash equilibrium is subgame perfect (Nash equilibrium) if the players’ strategies constitute a Nash equilibrium in every subgame.

(Selten 1965)

Note that every finite sequential game of complete information has at least one subgame perfect Nash equilibrium

We can find all subgame perfect NE using backward induction

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Backward Induction in dynamic games

of perfect information

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• Procedure:

– We start at the end of the trees

– first find the optimal actions of the last player to move

– then taking these actions as given, find the optimal actions of the second last player to move

– continue working backwards

• If in each decision node there is only one optimal action, this procedure leads to a Unique Subgame

Perfect Nash equilibrium

• Player 1 choose action a₁from the set A₁={Left, Right} • Player 2 observes a₁and choose an action a₂from the

set A₂={l, r}

• Payoffs are u₁(a₁, a₂) and u₂(a₁, a₂)

44 Player 1 Left Right Player 2 Player 2 l r l r U1(L,l) U2(L,l) U1(L,r) U2(L,r) U1(R,l) U2(R,l) U1(R,r) U2(R,r)

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• When Player 2 gets the move, she observes player’s 1 action a₁and faces the following problem

• Solving this problem, for each possible a₁ A₁, we get the best response of Player 2 to Player 1’s action.

• We denote it by R₂(a₁) , the reaction function of Player 2. • Player 1 can anticipate Player 2’s reaction, then Player

1’s problem is: 45 ) a , (a u Max ₂ ₁ ₂ A a2{ 2} )) (a R , (a u Max ₁ ₁ ₂ ₁ A a1{ 1}

• The backwards induction outcome is denoted by • It is different from the description of the equilibrium • To describe the Nash equilibrium we need to describe

the equilibrium strategies: – Action of Player 1 (Left or Right)

– Action of Player 2 after Left, action of player 2 after Right

46 )) (a R , (a*₁ ₂ *₁

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• Consider Player 2, the optimal action is accept

• Taking “accept” as given, we see that (9,1) is the optimal action for player 1

Player 1 (9,1) a r 9 1 0 0 (5,5) 5 5 Player 2 47

Example: Mini Ultimatum Game

The Ultimatum Game

• Proposer (Player 1) suggest (integer) split of a fixed pie, say £10. • Responder (Player 2) accepts the proposal or rejects (both receive 0) • There is no unique best response following (10,0), so we have two SPNE

Player 1 Player 2 (10,0) a r 10 0 0 0 (9,1) a r 9 1 0 0 a r (5,5) … 5 5 0 0 (0,10) … r a 0 10 0 0 48

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• First SPNE:

• Player 1 proposes (10, 0)

• Player 2 accepts in all of his decision nodes (a, a, a, a, a, a, a, a, a, a, a)

The Ultimatum Game

Second SPNE:

Player 1 proposes (9, 1)

Player 2 rejects after (10, 0) and accepts in all other decision nodes (r, a, a, a, a, a, a, a, a, a, a)

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Challenger Ready Unready Incumbent Fight 3 3 1 1 4 3 0 2

Acquiesce Acquiesce Fight

51 Three Nash equilibria

1. Ready, (A, F) 2. Unready, (A, A) 3. Unready, (F, A)

Only {Unready, (A, A)} is subgame perfect

Incumbent

(A, A) (A, F) (F, A) (F, F)

Challenger Ready 3, 3 3, 3 1, 1 1, 1

Unready 4, 3 0, 2 4.3 0, 2

Backward Induction in dynamic games of

imperfect information

• We start at the end of the trees

• first find the Nash equilibrium (NE) of the last subgame

• then taking this NE as given, find the NE in the second last subgame

• continue working backwards

If in each subgame there is only one NE, this procedure leads to a Unique Subgame Perfect Nash equilibrium

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Challenger Ready Unready Incumbent Fight 3 3 1 1 4 3 0 2 In 2 4

53 Challenger Incumbent Acquiesce Fight Challenger In Ready 3, 3 1,1 In Unready 4, 3 0, 2 Out Ready 2, 4 2, 4 Out Unready 2, 4 2, 4 Out 3 Nash equilibria:

• (In Unready) (Acquiesce) • (Out Unready) (Fight) • (Out ready) (Fight)

Subgame Perfect Nash equilibrium?

Start solving the last subgame

Challenger Ready Unready Incumbent Fight 3 3 1 1 4 3 0 2 Acquiesce Fight

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Incumbent

Acquiesce Fight Challenger Ready 3, 3 1,1

Unready 4, 3 0, 2 The normal form of the last subgame is:

Only one Nash equilibrium: (Unready) (Acquiesce)

Then the Challenger can predict that playing “In”

the outcome in the subgame will be the Nash equilibrium (Unready) (Acquiesce)

with payoff (4, 3) The reduced form is:

Then the best action the challenger can take is “in” Then the subgame perfect Nash equilibrium is (In Unready) (Acquiesce)

4 3 In 2 4 Challenger Out