GAME THEORY
DEFINITION: A formal analysis of situations where what is best for one agent to plan to do depends on
what other agents plan to do and at least the two parties involved are aware of this to some extent.
Game theory provides some general principles for thinking about strategic interactions, it can help an
oligopolist choose the course of action such as the best price to charge that maximizes its profit after
considering all possible reactions of its competitors
GAME: Any situation in which individuals must make strategic choices and in which the outcome will
depend on what each person chooses to do.
GAMES HAVE 4 BASIC CHARACTERISTICS
I.
PLAYERS
- Individuals who make decisions
- Each player’s goal is to maximize his objective function by choice of actions
II.
ACTION/MOVE/STRATEGY
- An action by player 1 denoted by 𝑎𝑖 is a choice he can make (e.g., set a high price, expand
plant size etc.)
III.
PAYOFF
- The expected utility he receives as a function of the strategies chosen by himself and other
players
- The utility he receives after all players and nature have picked their strategies and the game
has been played out.
IV.
INFORMATION SET
- His knowledge at a particular time of the values of different variables.
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TYPES OF GAMES
COOPERATIVE GAME- A game in which the players can negotiate binding contracts that allow
them to plan joint strategies.
NON-COOPERATIVE GAME- A game in which negotiation and enforcement of binding contracts
are not possible, players cannot come to binding agreements on how to play i.e. independent
decisions.
ZERO-SUM GAME- A game in which the sum of the payoffs of all the players is zero regardless of
whatever strategies they choose. A participant’s gain (or loss) of utility is exactly balanced by the
losses (or gains) of the utility of other participants.
NON-ZERO-SUM GAME- A game where one decision maker’s gain (or loss) does not necessarily
result in the other decision maker’s loss (or gain).
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STRATEGIES
DOMINANT STRATEGY- one that out-performs all the player’s strategies no matter what rivals do.
DOMINATED STRATEGY- one that is always out-performed by other strategies available to the
player.
PURE STRATEGY- One in which the player makes a specific choice or takes a specific action e.g.,
advertise or don’t advertise. These actions are taken with certainty.
MIXED STRATEGY- One in which a player makes a random choice among two or more possible
actions based on a set of chosen probability e.g., throwing a coin to select an action.
FUNDAMENTAL ASSUMPTIONS
RATIONALITY- Players are interested in maximizing their payoffs (profits, utilities).
COMMON KNOWLEDGE- all players know the structure of the game and that their opponents are
rational, that all players know the structure of the game and that their opponents are rational.
CLASSIFICATION OF GAMES
On the basis of;
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THE TIMING OF GAMES
- STATIC (SIMULTANEOUS) GAMES- each player moves once and without knowledge of the action
of her rivals
- DYNAMIC (SEQUENTIAL) GAMES- players move sequentially and have some idea about what
their rivals have done
- ONE SHOT GAME- played once and there is incentive to behave selfishly
- REPEATED GAMES- played more than once; there is scope for cooperative strategies to emerge.
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UNCERTAINITY ABOUT THE PAYOFFS OF RIVALS
- COMPLETE (SYMMETRIC) INFORMATION- Knowledge about other players is available to all
players.
- INCOMPLETE (ASYMMETRIC) INFORMATION – Not all players know each other’s utility
functions.
- PERFECT INFORMATION – Each player observes other players moves but lack some
information on other’s payoffs or structure of the game.
- IMPERFECT INFORMATION – Players are unaware of the actions chosen by other players.
CLASSES OF GAMES
1) GAMES OF COMPLETE INFORMATION
- Normal (or tabular) form games: Nash Equillibrium
- Extensive (or tree) form games: Subgame perfect Nash Equilibrium.
2) GAMES OF INCOMPLETE INFORMATION
- Normal form games: Bayesian Nash Equilibrium
- Extensive form games: Perfect Bayesian equilibrium.
3) STATIC GAMES OF COMPLETE INFORMATION
-Complete information means that players know the payoffs of their opponents.
-players have a single move and when a player moves he doesn’t know the actions taken
by her rivals.
NASH EQUILIBRIUM: A profile of strategies such that each player’s strategy is an optimal response to the
other players strategies. Each player’s NE strategy is a best response strategy to his opponents NE
strategies.
STRICTLY DOMINANT STRATEGY: Strategy that always provides greater utility to a player no matter what
the other player’s strategy is.
WEAKLY DOMINANT STRATEGY: Strategy that provides atleast the same utility for all the other player’s
strategies.
WEAK DOMINANCE: A strategy that is only weakly dominated cannot be ruled out based solely on
principles of rationality. Weakly dominated strategies could be dismissed if players believed there was at
least some positive probability that any strategies of their rivals could be chosen.
(STRICTLY) DOMINATED STRATEGY: “One strategy is strictly dominated for a player when there is
another strategy which leads to strictly better results (more payoff) no matter what combination of
strategies is used by the other players”.
WEAKLY DOMINATED STRATEGY: “One strategy is weakly dominated for a player if there is another
strategy which leads to results at least as good as those of the first one for any combination of strategies
of the other players and to strictly better results for some combination of strategies of the other players”.
In example 5, strategy s1 weakly dominates s2. Player 2 can conjecture that player 1 will play s1 and given
this conjecture the best she can do would be to play t2 . By following the criterion of weak dominance
(iterated deletion of weakly dominated strategies) the solution proposal would be (s1,t2 ).