2. Application of game theory
Game theory is the process of modeling
the strategic interaction between two or more
players in a situation containing set rules and
outcomes. While used in a number of
disciplines, game theory is most notably used as
a tool within the study of economics. The
economic application of game theory can be a
valuable tool to aide in the fundamental analysis
of industries, sectors and any strategic
interaction between two or more firms.
3. Game Theory Definitions
Any time we have a situation with two or
more players that involves known payouts or
quantifiable consequences, we can use game
theory to help determine the most likely
outcomes.
4. Game Theory Definitions
• Game: Any set of circumstances that has a result dependent on the
actions of two of more decision-makers (players)
• Players: A strategic decision-maker within the context of the game
• Strategy: A complete plan of action a player will take given the set
of circumstances that might arise within the game
• Payoff: The payout a player receives from arriving at a particular
outcome. The payout can be in any quantifiable form, from dollars
to utility.
• Information set: The information available at a given point in the
game. The term information set is most usually applied when the
game has a sequential component.
• Equilibrium: The point in a game where both players have made
their decisions and an outcome is reached.
8. Assumptions Used in Game Theory
• As with any concept in economics, there is the assumption
of rationality. There is also an assumption of maximization.
It is assumed that players within the game are rational and
will strive to maximize their payoffs in the game.
• When examining games that are already set up, it is
assumed on your behalf that the payouts listed include the
sum of all payoffs associated with that outcome. This will
exclude any "what if" questions that may arise.
• The number of players in a game can theoretically be
infinite, but most games will be put into the context of two
players. One of the simplest games is a sequential game
involving two players.
9. Solving Sequential Games Using Backwards
Induction
Below is a simple sequential game
between two players. The labels with Player 1
and Player 2 within them are the information
sets for players one or two, respectively. The
numbers in the parentheses at the bottom of
the tree are the payoffs at each respective point.
The game is also sequential, so Player 1 makes
the first decision (left or right) and Player 2
makes its decision after Player 1 (up or down).
10. Backwards induction, like all game theory, uses the assumptions of rationality
and maximization, meaning that Player 2 will maximize his payoff in any given
situation. At either information set we have two choices, four in all. By
eliminating the choices that Player 2 will not choose, we can narrow down
our tree. In this way, we will bold the lines that maximize the player's payoff
at the given information set.
11. After this reduction, Player 1 can maximize its
payoffs now that Player 2's choices are made known.
The result is an equilibrium found by backwards
induction of Player 1 choosing "right" and Player 2
choosing "up". Below is the solution to the game
with the equilibrium path bolded.
12. Prisoner’s Dilemma Basics
The prisoner’s dilemma scenario works as follows: Two suspects have
been apprehended for a crime and are now in separate rooms in a
police station, with no means of communicating with each other. The
prosecutor has separately told them the following:
• If you confess and agree to testify against the other suspect, who
does not confess, the charges against you will be dropped and you
will go scot-free.
• If you do not confess but the other suspect does, you will be
convicted and the prosecution will seek the maximum sentence of
three years.
• If both of you confess, you will both be sentenced to two years in
prison.
• If neither of you confess, you will both be charged with
misdemeanors and will be sentenced to one year in prison.
13. Zero-Sum Game
Zero-sum is a situation in game theory in which
one person’s gain is equivalent to another’s loss, so the
net change in wealth or benefit is zero. A zero-sum game
may have as few as two players, or millions of
participants.
Zero-sum games are found in game theory, but are
less common than non-zero sum games. Poker and
gambling are popular examples of zero-sum games since
the sum of the amounts won by some players equals the
combined losses of the others. Games like chess and
tennis, where there is one winner and one loser, are also
zero-sum games. In the financial markets, options and
futures are examples of zero-sum games, excluding
transaction costs. For every person who gains on a
contract, there is a counter-party who loses.
14. Example
One player’s gain is the other’s loss. The
payoffs for Players A and B are shown in the
table below, with the first numeral in cells (a)
through (d) representing Player A’s payoff, and
the second numeral Player B’s playoff. As can be
seen, the combined playoff for A and B in all
four cells is zero.
15. Zero-sum games are the opposite of win-
win situations – such as a trade agreement that
significantly increases trade between two
nations – or lose-lose situations, like war for
instance. In real life, however, things are not
always so clear-cut, and gains and losses are
often difficult to quantify.
16.
17.
18.
19. The dominant strategy
A strategy is dominant if, regardless of what
any other players do, the strategy earns a player a
larger payoff than any other. Hence, a strategy is
dominant if it is always better than any other
strategy, for any profile of other players' actions.
Depending on whether "better" is defined with
weak or strict inequalities, the strategy is
termed strictly dominant or weakly dominant. If
one strategy is dominant, than all others
are dominated. For example, in the prisoner's
dilemma, each player has a dominant strategy.
20. Strictly dominant strategy
A strategy is strictly dominant if, regardless
of what any other players do, the strategy earns
a player a strictly higher payoff than any other.
Hence, a strategy is strictly dominant if it is
always strictly better than any other strategy, for
any profile of other players' actions. If a player
has a strictly dominant strategy, than he or she
will always play it in equilibrium. Also, if one
strategy is strictly dominant, than all others
are dominated.
21. Weakly Dominant Strategy
A strategy is weakly dominant if, regardless of what
any other players do, the strategy earns a player a payoff at
least as high as any other strategy, and, the strategy earns
a strictly higher payoff for some profile of other players'
strategies. Hence, a strategy is weakly dominant if it is
always at least as good as any other strategy, for any profile
of other players' actions, and is strictly better for some
profile of others' strategies. If a player has a weakly
dominant strategy, than all others are weakly dominated.
If a strategy is always strictly better than all others
for all profiles of other players' strategies, than it is strictly
dominant.
23. A Nash equilibrium, named after John Nash, is a set of strategies, one for
each player, such that no player has incentive to unilaterally change her
action. Players are in equilibrium if a change in strategies by any one of them
would lead that player to earn less than if she remained with her current strategy.
For games in which players randomize (mixed strategies), the expected or average
payoff must be at least as large as that obtainable by any other strategy.
Nash Equilibrium
24. Mixed Strategy
A strategy consisting of possible moves and a probability distribution
(collection of weights) which corresponds to how frequently each move is to be
played. A player would only use a mixed strategy when she is indifferent
between several pure strategies, and when keeping the opponent guessing is
desirable - that is, when the opponent can benefit from knowing the next
move.
25. Minimax Strategy
In Minimax the two players are called maximizer and
minimizer. The maximizer tries to get the highest score possible
while the minimizer tries to do the opposite and get the lowest score
possible.
Minimax is a decision rule used to minimize the worst-case
potential loss; in other words, a player considers all of the best
opponent responses to his strategies, and selects the strategy such
that the opponent's best strategy gives a payoff as large as possible.
The name "minimax" comes from minimizing the loss
involved when the opponent selects the strategy that
gives maximum loss, and is useful in analyzing the first player's
decisions both when the players move sequentially and when the
players move simultaneously. In the latter case, minimax may give
a Nash equilibrium of the game if some additional conditions hold.
