1. TOPIC 1 What is a game? What does game theory study?
Strategic thinking: some examples
- Strategic voting in a committee
- “We had a flat tyre”
Our strategy for the course
- How to get good grades
- Organizational stuff

2. A game is any situation in which two or more decision makers (individuals or organizations) are aware that the result or payoff they obtain in the situation depends not only on her own decision but also on the decisions of the rest of decision makers.
A game is any situation of strategic interaction.
Gambling and sports: poker, chess, soccer, tennis…
War, divorce, relationship with parents, friends….
In the economy: oligopoly markets, allocation mechanisms such as auctions, bargaining between a buyer and a seller, labour relation, financial contract between a lender and a borrower….
And, also many situations that start out as markets governed by the impersonal forces of supply and demand turn into strategic interactions of two or just a few because of two possible reasons:
Mutual commitments.
Private information.
Rules of the game, known by the players.

3. Game Theory is the science of strategic thinking, that is, about interaction with others who are also doing similar thinking.
We assume as in economic theory that players are rational: they try to do the best they can given their information.
- in many examples and applications players try to maximize their material (or monetary) payoff. Selfish players.
- in other situations they are going to be motivated also by other goals such as relative payoffs, fairness…
Game theory adds another dimension, strategic thinking, that is, interaction with other equally rational players. A rational decision in a game must be based on “putting yourself in the other person´s shoes”.

4. STRATEGIC VOTING
A committee of a club has three members: A, B and C.
They have to make a decision on a prize. There are two candidates: Maria (M) and Jose (J).
The statutes of the club stipulate the following sequential procedure:
If there is more than one candidate the committee decides by simple majority in a secret votation who becomes the official candidate,
In a second secret votation and also by majority, they decide if the official candidate obtains the prize or nobody gets it.
Abstention is not allowed.

5. There are three possible results: Maria wins (M), Jose wins (J) or nobody wins (N).
The players (A, B and C) have well defined preferences on the results. In particular, A prefers M, in second place, N and the worst outcome for her is J. We denote this ordering as (M,N,J). Similarly, B´s ordering is (N,J,M) and C´s ordering is (J,M,N).
There is complete information: all players know the preferences of everybody (more precisely, all preferences are common knowledge among the players).
Paradox: how can it be better to vote for your worst alternative?

6. “We can´t take the exam, we had a flat tyre”
The professor agreed with the two friends and they had the exam on Tuesday instead of Monday.
They were placed in separate rooms and handed the test to each.
The first question, worth 1 point was easy. Each of them wrote a good answer and turned the page. It had just one question, worth 9 points.
It was: “Which tyre?”
Strategic lessons:
The professor may be an intelligent game player. You should look ahead to future moves in the game and then reason backward to calculate one´s best current action.
Can we independently produce a mutually consistent lie? Focal points.

7. A committee with three city councillors: Left (L), Center (C) and Right (R).
Three alternative welfare policies: generous (G), decreased (D) or maintain the status-quo or average (A).
L prefers G, followed by A and the worst outcome is D. We denote L´s ordering as (G,A,D). Similarly, C´s ordering is (A,D,G) and R´s ordering is (D,G,A).
The committee chair determines the specific order of voting (the agenda).
Votations are secret and by simple majority. Abstention is not allowed.

8. Suppose that L is the chair
Suppose that L is the chair. He proposes a first votation between A and D, followed by a second votation between the winner of the first and G.
There is complete information: all players know the preferences of everybody (more precisely, all preferences are common knowledge among the players).
Paradox: how can it be better to vote for your worst alternative?

9. FIRST-PRICE SEALED-BID AUCTION
Consider the owner of an indivisible good who decides to sell it through a public auction.
In a first-price sealed-bid auction the potential buyers are asked to issue their offers (bids) by, say, introducing them in a closed envelope. The seller opens these envelopes and the buyer with the highest bid gets the good and pays an amount equal to his bid to the seller.
Consider there are two buyers 1 and 2. The monetary valuation or reservation price of the good for buyer 1 is euros and this information is common knowledge. The monetary valuation for buyer 2 is private information of this buyer, but buyer 1 knows that it can take a value of 600 or 180 euros with equal probability.

10. There is a set of four closed envelopes containing each one the following amounts of euros:
(50, 100, 200, 400)
The referee gives to each of the two players one envelope from this set and announces them that one of the envelopes has twice the quantity of euros than the other.
You and your opponent have the opportunity to have a (secret) look to the content of your own envelope. You find out that yours has 100 euros.
The other player makes you the proposal of exchanging the envelopes. Would you accept?

11. Organizational Stuff
We will present the theory and the principles of strategic reasoning through cases: simple examples and economic applications.
No mathematical pre-requisites are needed.
Office: 3A09
Office hours: Tuesday, 11 – 13 h
Wednesday, – h
Thursday, – h
Web page: uv.es/olcina/

12. Nobel Prizes in Economics of Game Theory and applications. Laureates:
J. Nash, R. Selten and J. Harsanyi.
The Analysis of Equilibria in games
2001 J. Stiglitz, M. Spence and G. Akerloff.
Markets with asymmetric information
2002 D. Kahneman and V. Smith
Games and experimental economics
2005 T. Schelling and R. Aumann
Conflict and cooperation

13. TOPIC 2 SIMULTANEOUS GAMES: DOMINANCE
A simultaneous game is a game in which all players must move without knowledge of what their rivals have chosen to do.
Examples: “We had a flat tyre”, Sealed-bid auctions, Cournot duopoly, Rock-Paper-Scissors…
Strategic form of a game: players, sets of actions or strategies, utility or payoff functions.
The game matrix. Rows, columns and cells.

14. INCENTIVES TO WORK IN A TEAM
Joint production in a team, where individual costly work efforts are the inputs.
Effort is not observable or it is not verifiable (the information concerning effort is not admissible in court) and then it is not possible to enforce contracts where payoffs are conditioned to effort.
Assume that output and revenue are verifiable. The contract specifies a particular sharing rule of the total revenue or income generated by the team.

15. DOMINANCE: we say that an action of a player in a game dominates another action of this player if the former yields higher payoffs than the latter, for any possible combination of actions of her rivals (i.e. no matter what the others migth be doing). We will say that the second action is a dominated action.
DOMINANT ACTION: a player has a dominant action in a game if he has an action that dominates all the rest of her actions. That is, an action that yields a higher payoff than any other of her actions no matter what her rivals´actions are.
When a rational player has a dominant strategy or action, she wil use it.

16. EFFICIENCY
An outcome of a game is a vector of final payoffs, one for each player.
An outcome of a game is called efficient if there is no other outcome that is feasible and that yields a higher payoff to at least one player without giving a lower payoff to any player.

17. GAMES WITH A PRISONERS´DILEMMA STRUCTURE
Each player has a cooperative strategy, such that if everybody plays it, an efficient outcome is achieved.
But, each player has also a non-cooperative strategy (to defect from cooperation) that results in an inefficient outcome when everybody plays it. The non-cooperative action is a dominant action for a rational (and selfish) player (when the game is played once).
The payoff structure associated with the prisoners´dilemma arises in many quite varied strategic situations in economic, social, political, and even biological competitions.
The “rational” outcome in a prisoners´dilemma is a bad outcome for the players. What can players do to achieve the better outcome?

18. PRISONERS´DILEMMA
The police arrest two criminals guilty of murder. The police can prove illegal weapons´ possession , but cannot establish that either criminal committed murder unless at least one of them confesses.
The police put the criminals in separate rooms and asks them to confess. They explain the prisoners the consequences of their behaviour.
If both prisoners do not confess, then they wil get 2 years in prison each one because of weapons´possession.
If one confesses and the other does not, the former goes free and the latter gets 30 years of prison.
If both prisoners confess, each one gets 15 years of prison.
Would you confess or stay silent?

19. A weakly dominant action of a player is an action that yields her a higher or equal payoff than any other of her actions no matter what her rivals´actions are.
The best response function of a player.
It tells you the best choice (or choices) of this player, for each of the choices that the other players might be making.
Notice that to calculate any of these concepts a player only needs to know her own payoffs.

20. BEST RESPONSE ANALYSIS
The players´ best response functions provide an alternative method in two-player games to find dominance relations.
If a player has always a unique best response for any opponents´combination of actions, then this action is her dominat action.
If a player has an action that is always a best response (although not necessarilly unique), then it is her weakly dominant action.
If there is an action that it is never a best response, then it will be a dominated action (this result is valid taking into account the whole set of actions of the players, which means allowing for mixed actions).

21. SUCCESSIVE ELIMINATION OF DOMINATED STRATEGIES
A rational player will never play a dominated strategy. But if the game has complete information, then all players´ rationality and payoffs are common knowledge among them.
Players will use this information to mutually anticipate that dominated strategies will not be played and will remove them from consideration.
Removing dominated strategies reduces the size of the game and then the “new” game may have another dominated strategies. If this process keeps working until a unique outcome is reached, then the game is said to be dominance solvable.
Notice that each round or iteration in the process is justified by another step in the common knowledge´s chain.

22. PRIVATE PROVISION OF A PUBLIC GOOD
20 families in a residential area have to make a decision about the construction of a swimming pool. Its use is nonrival and nonexcludable among these families, so the swimming pool is a pure public good for them. Its cost is 100 monetary units. Some families do not value the swimming pool at all,that is, it would report an utility of zero for any of them. But other families assign a positive value to the good and would obtain an utility of 20, each one, in case it is constructed.
The collective decision is made with the following procedure: each family votes Yes or No in a closed envelope; only if at least 10 families vote yes, the swimming pool is constructed, paying proportionally the cost 100 those who voted yes.
What would happen if the families do not know who assigns positive value to the swimming pool and who does not?
What happens if it were common knowledge that ten families assign positive value to the swimming pool and the rest do not?

23. GUESS HALF THE AVERAGE
Write down in a card your name and a natural number between 0 and 100.
The person whose choice is closest to half of the average is the winner.

24. Even if everyone chooses 100, half of the average can never exceed 50; so, for each player, any choice above 50 is dominated by 50.
But all players should rationally figure this out, so the average can never exceed 50 and half of it can never exceed 25, and so any choice above 25 is dominated by 25.
The iteration goes on until only 0 is left.

25. EVALUATION BY RELATIVE PERFORMANCE
Imagine that I announce the following grading for this course: no matter how well you do in absolute terms in the exam, only 50% of the students will pass and the other 50% will not. Will you work hard or not in this situation?
A firm evaluates each salesperson on a relative standard, i.e. comparing him to his peers. Explain how this creates a prisoners´dilemma for employees.

26. Why do some sports leagues pay their athletes so much?
If all basketball teams offered much lower salaries, they would still attract the top talent since this talent would have no other place to go. But individual teams face a prisoners´dilemma with respect to their players´salaries.
A solution: a salary cap
Why do rational companies need to pay astronomical sums to attract the best bosses (CEOs)?

27. A GENERALISED PRISONERS´DILEMMA
Four individuals have to decide simultaneously their contribution to a public good. Individual contributions xi can adopt only three values: 0, 100 or 200. The payoff function of each player, given a combination of contributions, would be:
ui(x) = 200 – xi + 0,5(x1 + x2 + x3 + x4), where 0,5 is the marginal return of the public good.
a) What would be the individuals´contributions?
b) Calculate the efficient outcome in this situation.

28. TO FINANCE OR NOT
Two companies sell the same model of car. They have to decide simultaneously whether or not to provide the possibility of financing the purchase.
The cost of providing financing is 3 millions of euros, but it increases the total demand and allows you to capture a higher market share provided the other firm does not finance. In particular, without financing, the market demand is 12 millions and with financing 14 millions. If one firm finances and the other does not, the former captures 75% of the market.
If both firms choose the same action, then they share equally the total demand.
Construct the game matrix of this situation and make a prediction.

29. TOPIC 3 HOW DO WE PLAY IN REAL-LIFE GAMES WITHOUT DOMINATED ACTIONS?
Expert´s advice or recommendation (nevertheless, it is only a recommendation, that is, we keep our strategic freedom).
Conventions or social norms (but you are free to ignore the norm).
Learning and experimentation: we learn after playing many times similar games against different opponents through trial and error.
If, for any of these motives, it appears a stable way to play the game, predicted and expected by everybody, it (a combination of strategies) has to be a Nash equilibrium.
For instance: a minimal property that an expert´s recommendation has to satisfy is that it must be in the self-interest of any player to follow the recommendation if the other players are going to follow it.

30. TOPIC 3 NASH EQUILIBRIUM
A Nash equilibrium (NE) of a game is a combination of strategies, one for each player, such that the strategy of any player is a best response to her opponents´strategies.
In a NE, no player regrets his strategy, given everyone else´s strategies (although, you are not necessarilly happy with the other players´strategies!).
In a NE, no player has a profitable unilateral deviation, that is, no player can get a better payoff by unilaterally switching to some other available strategy while all the other players adhere to the strategies specified for them in the list.
How do we compute the NE of a game? Using the best response functions of the players. NE are the “intersections” of these functions.
How do we check if a particular combination of actions is a NE? Finding if there is any player with a profitable unilateral deviation.

31. NASH EQUILIBRIUM AND DOMINANCE.
a) In a game in which all players have a dominant action, the combination of actions in which everybody plays his dominant action is the unique NE of the game.
b) If a game is dominance-solvable, the combination of actions that survives the iterative process of elimination of dominated actions is the unique NE of the game.
c) A player never plays a dominated action in a NE. Nevertheless, in some games, there might be a NE where some players play a weakly dominated action.

32. A LOCATION PROBLEM
Two ice-cream vendors, that sell a homogeneous good, have to decide simultaneously their respective locations in a one-hundred meter beach. We will represent formally the beach with the interval [0,1]. Consumers are distributed in this space according to a uniform distribution. They prefer to buy the ice cream from the closest vendor (there is a transportation cost). Which is the NE of this game?

33. The best response to any of your opponent´s location is to place yourself beside him where there is a larger extension of beach.
Therefore, there will not be an equilibrium if there is more extension of beach on one side than on the other.
The unique NE is that both players place themselves in the middle of the beach.

34. A DUOPOLY WITH PRICE COMPETITION
Two firms that produce a homogeneous good have to set prices simultaneously. Firms have the same constant unitary costs c >0. The market demand is given by a function D(p) with negative slope, where p is the price. Consumers have perfect information and no transportation costs, so they will buy from the firm that sets the lowest price. In case both firms set the same price, each one will capture half of the market demand. Calculate the NE.

35. TEAM PRODUCTION.
Four individuals jointly produce a good. The level of output depends on the non-verifiable work effort made by each of the individuals. Work effort can take three values 0, 1 and 2. The individual cost of effort is given by the functions c(ei) = ei , i = 1,2,3,4. The total revenue or income generated by the team is given by the function: R = 2.(e1 + e2 + e3 + e4).
Assume initially that the members of this team write down a contract where they agree on sharing equally the (verifiable) total revenue.
Find the Nash equilibria of this simultaneous game and discuss its efficiency.
Is it possible to obtain efficiency with another contract (with other distribution rule)?
Suppose now that there is an owner of the firm who does not participate in the production. He proposes the following contract to each of the four workers: If total revenue is greater or equal than 16, the wage will be 3 and if revenue is smaller than 16, the wage will be 0,5. Find the Nash equilibria.

36. It is not possible to obtain the efficient levels of effort with sharing rules that always distribute all the revenue generated by the team. The reason is that the rule should estipulate, in our example, that each worker will obtain at least half of each additional euro generated by his effort. This is impossible to be satisfied for the four workers at the same time.
Nevertheless, it is possible to obtain the efficient result with a sharing rule that only distributes the whole revenue if a particular level of revenue is reached.
Problem: contracts can be renegotiated. Who implements the punishment? Will the group implement it?
A possible solution is the existence of a legal residual claimant for this revenue: the owner of the firm.