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Solving Games Nash Equilibrium

- Solution Concept a methodology for predicting

player behavior. - Nash Equilibrium - a collection of strategies one

for each player, such that every players

strategy is optimal given that the other players

use their equilibrium strategy.

Backward Induction

- 1. Start at terminal nodes of the game, and trace

back to its parent. - 2. Find optimal decision for that player at that

decision node by comparing payoffs the player

receives from each terminal node. Record this

choice, it is part of the players optimal

strategy. - 3. Prune from the game tree all the branches that

originated from 1. Attach to each of these new

terminal nodes the payoffs received when the

optimal action is taken at this node. - 4. A new game tree exists and is smaller than the

original, if there are no decision nodes, you

are done. - 5. If there are decision nodes, apply 1 thru 4

until no decision nodes are left. - 6. For each player collect the optimal decisions

at every node. This collection constitutes that

players optimal strategy in the game.

Backwards Induction, cont.

- Review the software game in 5.1 Use backwards

induction to determine both players optimal

decisions. - Is this a Nash Equilibrium Yes.
- Why All backwards induction solutions are Nash

equilibrium.

Threats Credible vs. Incredible

- Inability to make enforceable commitment, not no

communication drive the above results. - Suppose Microcorp tells Macrosoft it will always

clone the software no matter what advertising

campaign Macrosoft employs. This is a Threat. - A threat occurs when one player attempts to get

other players to believe it will employ a

specific strategy. - Given this threat, Macrosoft would be wise to

ignore this threat, because it is not credible. - A threat by a player is not credible unless it is

in the players own interest to carry out the

threat when given the option. Threats that are

not credible are ignored.

Threats, cont.

- A binding threat is one in which a player can not

back down. - FIGURE 5.2 gives a binding threat game tree.

Note Idle is dropped for simplicity. - What is Macrosofts optimal strategyMA only if

Microcorp threatens EMA, EWOM and WOM for

all other strategies - Note There is a relationship between credible

threats and Nash equilibrium. If Macrosoft

believes that Microcorp will employ ENTERMA,

ENTERWOM, then Macrosoft should play WOM. It is

also true that if Microcorp believes that

Macrosoft plays WOM it should play ENTERMA,

ENTERWOM, which is a Nash Equilibrium. Yet it

is a Nash equilibrium that relies on an

incredible threat.

Subgames

- A subgame is essentially a smaller game within a

larger game with two special properties. - Once the players begin playing a subgame, they

continue playing the subgame for the rest of the

game. - The players all know when they are playing the

subgame. - A subgame has one initial node, the subroot.

Subgames, cont.

- The subgame Gs of the game GT is a game

constructed as follows - Gs has the same players as GT, although some of

these players may not make any moves in Gs - The initial node of Gs is a subroot of GT, and

the game tree of Gs consists of this subroot, all

its successor nodes, and the branches between

them. - The information sets of Gs consist of those

information sets of GT that contain a decision

node of Gs. - The payoffs of each player at the terminal nodes

of Gs are identical to the payoffs in GT at the

same terminal nodes. - These four conditions say that the set of

players, order of play, set of possible actions

and the information sets in the original game are

preserved in the subgame.

Subgames, cont.

- This definition implies that every game is a

subgame of itself. - FIGURE 5.3 How many subgames exist Four.
- Playing any subgame is common knowledge to the

players. Thus rational behavior for the players

in the overall game should also appear rational

when viewed from the perspective of this subgame. - WOM, ENTERMA, ENTERWOM, STAY OUTIDLE is

a Nash equilibrium that involves an incredible

threat by Microcorp to enter no matter what

advertising strategy Macrosoft uses. - The subgame in Figure 5.3 at D2 has only one Nash

equilibrium which Microcorp chooses Stay Out. - Yet the strategy profile selects Microcorp to

enter. - It is this non-optimality in the subgame which

makes the threat incredible.

Subgame-Perfect Equilibrium

- A Nash equilibrium strategy that remains a Nash

equilibrium when applied to any subgame is

subgame-perfect. - A strategy profile is a subgame-perfect

equilibrium of a game G if this strategy profile

is also a Nash equilibrium for every subgame of

G. - In games of perfect information, subgame-perfect

equilibria are those found via backwards

induction. - Backwards induction eliminates all the incredible

threats, and all actions are a Nash equilibrium,

and subgame-perfect equilibrium. - REVIEW THE SOFTWARE GAME WITH DIFFERENT PAYOFFS.

Games with Perfect Information

- Games with Uncertain Outcomes involve a new

player, Nature. When we come across an example

of a game with Nature, we will return to the

details of this type of game. - Games with Perfect Information and Continuous

Strategies - How much output should I produce
- How much advertising should I purchase
- The Stackleberg Duopoly

Chapter Eleven Nash Equilibrium II

- Modeling Simultaneous-Move Games
- Review Figure 11.1
- The model appears sequential.
- Since the decision nodes for Rusty are in the

same information set, the model is simultaneous.

- Remember, simultaneous is not a reference to

chronological time, but rather the fact that each

player does not know the other players move

before she/he makes her/his move. - Hence, Figure 11.2 is equivalent to Figure 11.1

Dominant and Dominated Strategies

- Payoff matrix a matrix that displays the

payoffs to each player for every possible

combination of strategies the players could

choose. Review Table 11.4 - Dominant Strategy a strategy that is always

strictly better than every other strategy for

that player regardless of the strategies chosen

by the other players. - Dominated Strategy a strategy that is always

strictly worse than some other strategy for that

player regardless of the strategies chosen by the

other players.

Weakly Dominate Strategies

- Weakly dominant strategy - a strategy that is

always equal to or better than every other

strategy for that player regardless of the

strategies chosen by the other players. - Weakly Dominated Strategy a strategy that is

always equal to or worse than some other strategy

for that player regardless of the strategies

chosen by the other players.

Prisoners Dilemma

- Scenario Two people are arrested for a crime
- The elements of the game
- The players Prisoner One, Prisoner Two
- The strategies Confess, Dont Confess
- The payoffs
- Are on the following(payoffs read 1,2)

Prisoners Dilemma, cont.

- Prisoner 2
- Confess Dont Confess
- Confess 2 years, 2 years 0 years, 10 years
- Prisoner 1
- Dont Confess 10 year, 0 years 5 years, 5

years - Dominant strategy equilibrium In this game, the

dominant strategy for each prisoner is to

confess. So the outcome of the game is that they

each get two years. - This illustrates the prisoners dilemma games in

which the equilibrium of the game is not the

outcome the players would choose if they could

perfectly cooperate.

The Advertising Game

- Scenario Two firms are determining how much to

advertise. - The elements of the game
- The players Firm 1, Firm 2
- The strategies
- High advertising, low advertising

Advertising Game, Cont.

- The payoffs Are as follows (payoffs read 1,2)
- Firm 2
- High Low
- High 40,40 100, 10
- Firm 1
- Low 10, 100 60,60
- Dominant strategy equilibrium In this game, the

dominant strategy for firm 1 and firm 2 is high.

So the outcome of the game is 40,40. - Again, this is an example of the prisoners

dilemma. The equilibrium of the game is not the

outcome the players would choose if they could

cooperate.

More Prisoner Dilemmas

- Industrial Organization Examples
- Cruise Ship Lines and the move towards glorious

excess. Royal Caribbean offers a cruise with an

18 hole miniature golf course. Princess Cruises

has a ship with three lounges, a wedding chapel,

and a virtual reality theater. - Owners of professional sports teams and the

bidding on professional athletes. - Non-IO Examples
- Politicians and spending on campaigns.
- Worker effort in teams. The incentive exists to

shirk, a strategy that if followed by all

workers, reduces the productivity of the team.

More on shirking later.

Iterated Dominant Strategies

- What if a dominant strategy does not exist
- We can still solve the game by iterating towards

a solution. - The solution is reached by eliminating all

strategies that are strictly dominated.

Example of Iterated Dominance

- Down is Firm 1, Across is Firm 2

Alternative Solution Strategies

- Nash Equilibrium - a strategy combination in

which no player has an incentive to change his

strategy, holding constant the strategies of the

other players. - Joint Profit Maximization This is the objective

of a cartel. - Cut-Throat A strategy where one seeks to

minimize the return to her/his opponent. - How does the previous game change when we change

the objectives of the players - This is one of the advantages of game theory. We

do not have to assume profit maximization. We

still need to be able to identify the objectives

of the players.

A Lack of Dominance

- Down is Player 1, Across is Player 2

A Lack of Dominance, cont.

- Given these payoffs, is there a dominant or

dominated strategy - If 1 chooses A, 2 will choose C
- If 1 chooses B, 2 will choose B
- If 1 chooses C, 2 will choose A
- Likewise
- If 2 chooses A, 1 will choose A
- If 2 chooses B, 1 will choose B
- If 2 chooses C, 1 will choose C
- Therefore, no dominant or dominated strategy

exists. Is there a Nash equilibrium - What if player 1 chose C, and player 2 chose A,

is this a Nash Equilibrium - No, if player 2 chose A, player 1 would want A.
- Only when both choose B, or both happy with the

choice, therefore this is a Nash equilibrium.

Mixed Strategy

- Pure Strategy is a rule that tells the player

what action to take at each information set in

the game. - Mixed strategy allows players to choose randomly

between the actions available to the player at

every information set. Thus a player consists of

a probability distribution over the set of pure

strategies. - Examples of mixed strategy games
- Play calling in sports
- To shirk or not to shirk

The Shirking Game

- Scenario A worker is hired but does not wish to

work. The firm will not pay the worker if there

is no work, but the firm cannot directly observe

the workers effort level or output. - Players The worker, the firm
- Strategy Work or not work, monitor or not

monitor - Payoffs Work pays 100, but the workers

reservation wage is 40 - Worker can produce 200 in revenue, but it costs

80 to monitor

The Shirking Game, Cont.

- There is no dominant strategy, or iterated

dominant strategy. - There is also no clear Nash Equilibrium. In

other words, no combination of actions makes both

sides happy given what the other side has chosen. - There are many mixed strategies. The worker could

work with probability (p) of 0.7, 0.6. 0.25,

etc... The same is true for the firm. Which

mixed strategy should they choose - If the worker is most likely to shirk, the firm

should monitor. Likewise, if the firm is more

likely to monitor, the worker should work. In

any scenario, no Nash equilibrium will be found.

The key is to find a strategy that makes the

opponent indifferent to his/her potential

choices. - A person is indifferent when the expected return

from action A equals the expected return form

action B.

Solving the Shirking Game

- How much should the firm monitor
- E(work) 60p 60(1-p) 60
- E(shirk) 0p 100(1-p) 100 - 100p
- 100 - 100p 60
- 40 100p
- p .40
- The worker is indifferent when the probability of

monitoring is 40 and the probability of not

monitoring is 60. - How much should the worker work
- E(monitor) 20p -80(1-p) 100p - 80
- E(Not monitor) 100p -100(1-p) 200p - 100
- 100p -80 200p - 100
- 20 100p
- p .2
- The firm is indifferent when the probability of

working is 20 and the probability of not working

is 80. - How does the cost of monitoring and the workers

reservation wage impact behavior

Existence of Nash Equilibrium

- Every game with a finite number of players, each

of whom has a finite number of pure strategies,

possesses at least one Nash equilibrium, possibly

in mixed strategies - Final Note If the players have continuous

strategies (as opposed to finite strategies) a

pure strategy can be found with a reaction

function.

The Football Game

- Scenario A game has come down to a final play.

The 49ers are on the 2 yard line with 5 seconds

to go. The current score is 20-16, with the

Raiders in the lead. The 49ers have two choices,

run or pass. The Raiders have two choices, defend

against the run or defend against the passes. - Players 49ers, Raiders
- Strategy Play Pass or Run, Defend Pass or Run
- Payoffs Probability of success given choices

The Football Game, cont.

- There is no dominant strategy, or iterated

dominant strategy. - There is also no clear Nash Equilibrium. In

other words, no combination of actions makes both

sides happy given what the other side has chosen. - Hence this is a mixed strategy game.
- Remember, a person is indifferent when the

expected return from action A equals the expected

return form action B.

Solving the Football Game

- Should the 49ers run or pass
- E(D run) 70p 20(1-p) 2050p
- E(D pass) 30p 80(1-p) 8050p
- 20 50p 80 50p
- 100p 60
- p .60
- The Raiders are indifferent when the 49ers run

60 and pass 40 of the time. - Should the Raiders defend the run or pass
- E(run) 30p 70(1-p) 70 40p
- E(pass) 80p 20(1-p) 60p 20
- 70 40p 60p 20
- 50 100p
- p .5
- The 49ers are indifferent when the Raiders defend

the run 50 of the time.

Who will win the game

- The probability that the 49ers will win the game

the Nash Equilibrium strategies are adopted

equals - 0.6 0.5 30 0.6 0.5 70 0.4 0.5 80

0.4 0.5 20 50 - The 49ers have a 50 chance of winning this game

when each team adopts their equilibrium

strategies.

The Football Game, new payoffs.

- How does changing the expected payoffs alter the

probabilities that each team will take each

action - The 49ers have a very good chance of scoring if

they pass, and the Raiders play run defense. - Outcome of the game
- 49ers will run with a probability of 4/7
- Raiders will play the run with a probability of

2/7

Who will win the game now

- The probability that the 49ers will win the game

the Nash Equilibrium strategies are adopted

equals - 4/7 2/7 40 3/7 2/7 90 4/7 5/7 70

3/7 5/7 50 61.4 - The 49ers have a 61.4 chance of winning this

game when each team adopts their equilibrium

strategies.

The Voting Game

- Non-intuitive game theory voting paradoxes
- Scenario Three economist need to decide how much

math to require for economics majors. The

options are - 1) require no math
- 2) require one semester calculus
- 3) require two semesters calculus
- Preferences of each professor Dr. Vaitheswaran

(V) LMH - Dr. Berri (B) MHL
- Dr. Wu (W) HLM
- V is the chair of the committee, and V has the

power to break any ties. Voting will be done

simultaneously by secret ballot. - Naive voting Professors ignore that it is a game

and simply vote their preferences. - Outcome V breaks the tie as the chair and the

students at Coe have no math requirement.

The Voting Game, Cont.

- On the left are the outcome of the game, given

each possible combination of votes for B and W,

and each vote for V. - The outcome in bold is the preferred outcome for

V. - V has a weakly dominant strategy (L). In three

instances, Vs vote would be irrelevant,

therefore V would not have a preference. In

every other instance, V would maximize his

utility by voting (L). From this we can

conclude that V will vote (L).

The Voting Game, Cont.

- Voting for (L) is weakly dominated by (H) and

(M), since this is the least of Bs preferences. - Therefore, B will not choose (L), and we can

eliminate this option.

The Voting Game, Cont.

- For W, (M) is weakly dominated by (H) and (L).

Given this, W will choose (H) in every instance,

so (H) is weakly dominant. - The outcome of the game then is as follows
- V will vote L
- W will vote H
- B will vote H
- The students at Coe will thus have a high math

requirement, exactly the opposite - of what the chair wants.

The Good, The Bad, and the Ugly

- Scenario Three gunfighters in a gun fight. The

winner gets the gold. - Players Good is the fastest, Bad is the second

fastest, and Ugly is the slowest at firing a gun. - Each gunfighter only gets one shot, if he is not

killed by a faster person. The winner gets the

gold. If two people survive, the two agree to

split the gold. - All three gunfighters know the skill level of

their opponents. - Potential Actions Shoot at one of the remaining

players.

The Good, The Bad, and the Ugly, cont.

- Ugly has a dominant strategy. If Ugly aims at

Good, he is always better off than when he aims

at Bad. - Bad has the same dominant strategy. Aiming at

Good results in a higher payoff than aiming at

Ugly. - Hence, in this game, the fastest gunfighter is

killed.

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