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318 Game Theory notes
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GAME THEORYHOW TO THINK ABOUT STRATEGIC
GAMES
WHAT IS GAME THEORY?
- Game theory is the science of rational behaviour in strategic situations
- This means that the outcome (“payoff”) for an agent depends on the actions of other agents.
- Further, there must be mutual awareness that each agent’s actions influence the others
- Whenever you have a situation with an externality, there’s usually game theory
What’s the relationship between game theory and neo-classical economic theory?
- The first theoretical ideas about how economies work were perfectly competitive markets
- We used to have to make theories around the fact that we did not have the ability to study
strategic interaction
- What about imperfectly competitive markets?
• Eg a monopoly
• Not really strategic action
• The monopolist does not really have to worry about how any of the individuals will behave
Conventional Game Theoretic
Microeconomics Analysis
Prices taken as Individuals Agents’ decisions
given make decisions impact on payoffs
of other agents
Individual Model-Based
agents cannot Agents’ decisions
Rational depend on other
influence
prices Optimization agents’ decisions
But we do not necessarily judge a model for its ability to predict accurately. We judge a model on
its ability to explain behaviour
BRIEF HISTORY
1838: Cournot model
- A model that has two firms choosing goods to produce
- What’s impressive about his model, was that it was so early on and we still see it as a very
valuable contribution
1928: On the theory of parlour Games (John von Neumann)
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,- JvN helped design the American atomic bomb
- Basically invented game and decision theory
- His paper was dealing with games people played for fun such as chess and checkers
1944: Theory of Games and Economic Behaviour (John von Neumann and Oskar von
Morgenstern)
- JvR teamed up with an actual economist, Morgenstern. This is where game theory in economics
actually started
1950: John Nash’s PhD thesis
DECISIONS VS GAMES
For the interaction to become a strategic game, we need something more—namely, the
participants’ mutual awareness of this cross-effect. What the other person does affects you; if you
know this, you can react to his actions, or take advance actions to forestall the bad effects his
future actions may have on you and to facilitate any good effects, or even take advance actions so
as to alter his future reactions to your advantage. If you know that the other person knows that
what you do affects him, you know that he will be taking similar actions.
It is this mutual awareness of the cross-effects of actions and the actions taken as a result of this
awareness that constitute the most interesting aspects of strategy.
If Robert E. Lee (who ordered Pickett to lead the ill-fated Pickett’s Charge at the battle of
Gettysburg) had thought that the Yankees had been weakened by his earlier artillery barrage to the
point that they no longer had any ability to resist, his choice to attack would have been a decision;
if he was aware that the Yankees were prepared and waiting for his attack, then the choice
became a part of a (deadly) game.
Decision theory: u = f(x)
Game theory: u1 = f(x1,x2)
- And potentially u1 = f(x1,x2(x1))
Game theory has a much greater scope. Many situations that start out as impersonal markets with
thousands of participants turn into strategic interactions of two or just a few. This happens for one
of two broad classes of reasons—mutual commitments or private information.
CLASSIFYING GAMES
1. Are the moves in the game sequential or simultaneous?
• Moves in chess are sequential: white first, then black
• You think “if I do this, how will my opponent respond?”
• In simultaneous games, you make your moves at the same time
2. Are the players’ interests in total conflict or is there some commonality?
• In simple games such as chess there is a winner and a loser. One player’s winnings are the
others’ losses, so the total is 0. This is why such situations are called zero-sum games
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, • More generally, the idea is that the players’ interests are in complete conflict. Such conflict
arises when players are dividing up any fixed amount of possible gain, whether it be
measured in yards, dollars, acres, or scoops of ice cream. Because the available gain need
not always be exactly 0, the term constant-sum game is often substituted for zero-sum game
• Most economic and social games are not zero-sum. Even wars and strikes are not zero-sum
(except maybe nuclear war)
3. Is the game played once or repeatedly, and with the same or changing opponents?
• A game played just once is in some respects simpler and in others more complicated than one
that includes many interactions. You can think about a oneshot game without worrying about
its repercussions on other games you might play in the future against the same person or
against others who might hear of your actions in this one. Therefore actions in one-shot
games are more likely to be unscrupulous or ruthless. For example, an automobile repair shop
is much
- Therefore in one-shot games, secrecy or surprise is likely to be an important component of
good strategy.
• Games with ongoing relationships require the opposite considerations. You have an
opportunity to build a reputation (for toughness, fairness, honesty, reliability, and so forth,
depending on the circumstances) and to find out more about your opponent. The players
together can better exploit mutually beneficial prospects by arranging to divide the spoils over
time (taking turns to “win”) or to punish a cheater in future plays (an eye for an eye or tit-for-
tat).
4. Do the players have full or equal information?
- This type of information problem arises because of the player’s uncertainty about relevant
variables, both internal and external to the game.
- When different players have different information, the manipulation of information becomes a
game itself
5. Are the rules of the game fixed or manipulable?
- The rules of chess, card games, or sports are given, and every player must follow them, no
matter how arbitrary or strange they seem. But in games of business, politics, and ordinary life,
the players can make their own rules to a greater or lesser extent.
- In such situations, the real game is the “pregame” where rules are made, and your strategic skill
must be deployed at that point. The actual playing out of the subsequent game can be more
mechanical; you could even delegate it to someone else. However, if you “sleep” through the
pregame, you might find that you have lost the game before it ever began. For many years,
American firms ignored the rise of foreign competition in just this way and ultimately paid the
price.
6. Are agreements to cooperate enforceable?
- Agreements to cooperate can succeed if all players act immediately and in the presence of the
whole group, but agreements with such immediate implementation are quite rare
- Game theory uses a special terminology to capture the distinction between situations in which
agreements are enforceable and those in which they are not. Games in which joint-action
agreements are enforceable are called cooperative games; those in which such enforcement is
not possible, and individual participants must be allowed to act in their own interests, are called
noncooperative games.
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, TERMINOLOGY
1. Strategies
• Simply the choices available to the players
2. Payoffs
• The number associated with each possible outcome
3. Rationality
• Much of game theory assumes that players are perfect calculators and flawless followers of
their best strategies
• This does not mean that players are selfish
4. Common knowledge of rules
• Game theory cannot properly analyse a situation where one player does not know whether
another player is participating in the game
5. Equilibrium
• Each player is using the strategy that is the best response to the strategies of the other
players.
6. Dynamics and evolutionary games
• The evolutionary approach to games does just this. It is derived from the idea of evolution in
biology. Any individual animal’s genes strongly influence its behaviour. Some behaviours
succeed better in the prevailing environment, in the sense that the animals exhibiting those
behaviours are more likely to reproduce successfully
• The object of study is the dynamics of this process
THE USES OF GAME THEORY
1. Explanation
2. Prediction
3. Advice or prescription
Examples
- Chess
• Zero sum game
• Has well defined rules and moves
• We know who the players are
- Rock paper scissors
• Zero sum game
- Oligopoly pricing
• Not a zero sum game
- The GPA rat race
• Note the importance of cooperation
- “We can’t take the exam because we had a flat tire”
• Note the important element of looking ahead and reasoning backwards
- Economic policy
- Bribes
- Wage bargaining
- Auctions
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