© 1999-2020, Douglas A.Ruby (06-05-2020)

A fairly recent development in microeconomics has been the introduction of Game Theory as an analytic tool to understand the behavior of individual economic agents. This particular form of modeling takes into account the use of strategy rather than marginal analysis to support the decision-making process. It is still assumed that economic agents engage in optimizing behavior. The difference is in the use of information of potential outcomes based on choices made by other agents.

Games can be developed under conditions of complete, partial, or asymmetric information among the players. The final solution or potential equilibrium of a game depends on actions and reactions of the players--reactions that may change if the game is repeated rather than played only once.

The basic tool of game theory is the payoff matrix. This matrix represents known payoffs to individuals (players) in a strategic situation given choices made by other individuals in that same situation. For example :

Agent A: / Agent B: |
Choice I |
Choice II |

Choice I |
a_{1,1} ,
b_{1,1} |
a_{1,2},
b_{1,2} |

Choice II |
a_{2,1} ,
b_{2,1} |
a_{2,2},
b_{2,2} |

The entries 'a_{ij},b_{ij}
' represent numeric payoff to Agent **A** and Agent **B**
respectively. If possible, choices made by each player
will be independent of the actions of the other
player--one player is ignorant of the choice to be made
by the other. However, if for Agent **A**: a,_{1,j}
> a_{2,j} for all values of 'j' then this person will
always choose **Choice I.** This would represent a dominant strategy for Agent
**A** The same could be true for Agent **B**: if
b_{i,1}, > b_{i,2} for all values of 'i', **Choice I**
would be a dominant strategy for this second player.

When both players have a dominant strategy, an equilibrium exists in the model as defined by the cell corresponding to the optimal choices of both players. Even if only one player has a dominant strategy, an equilibrium can be determined given that the other player will react to this optimal choice made by the former.

For a numeric example:

Firm A: / Firm B | Choice I |
Choice II |

Choice I |
1,2 | 3,1 |

Choice II |
2,5 | 5,4 |

The question is: *What choice will each firm actually make?*

IfFirm BchoosesIthenFirm Awill chooseIIsince the payoff to firm A is higher ($2.00 vs. $1.00).

IfFirm BchoosesII,Firm Awill still chooseII.

So, independent of **Firm B**'s choices, **Firm A** will always make choice **II**-- its *dominant strategy*.

FromFirm B's point of view, ifFirm AchoosesI,Firm Bwill chooseI.

IfFirm AchoosesII,Firm Bwill still chooseI.

**Firm B** will always make choice** I**--the *dominant strategy*.

The payoffs in the lower left-hand corner (**A = II, B = I**)
represent *an equilibrium*. The presumed strategy was the maximization
of the individual payoffs.

It will not always be the case that an equilibrium under this maximization strategy will exist. An example is the payoff matrix given below:

Firm A: / Firm B | Choice I |
Choice II |

Choice I |
1,2 | 4,0 |

Choice II |
3,1 | 0,3 |

Under maximization no equilibrium point exists:

- If
**Firm A**chooses**I**,**Firm B**will choose**I**. - If
**Firm A**chooses**II**,**Firm B**will choose**II** - If
**Firm B**chooses**I**,**Firm A**will choose**II** - If
**Firm B**chooses**II**,**Firm A**will choose**I**

Firm A's choice *depends* on the choice made by
firm B and vice-versa. Instead a different strategy may
be employed. This new strategy is to make *the best of
the worse-case scenario*. In the case of **Firm A**,
choosing** I** would mean a minimum payoff of $1.00
(if **B** chose **I**). If **Firm A** chose **II**
then the minimum payoff would be $0.00 (if **B** chose
**II**). So **Firm A**, taking a cautious approach
would always choose** I**. **Firm B**, using the
same strategy would also always choose **I**.
This cautious strategy is known as a maxi-min strategy or *maximizing
the minimum-possible *payoff.

In the above games efficiency may be determined via
the notion of Pareto-Optimality
that is, given the point of equilibrium, is it possible *to
make one person better off with-out harming the other
person?* If not then the solution is Pareto Optimal or Pareto
Efficient. If the equilibrium is not Pareto Optimal then a better
outcome exists with respect to the goals of the players involved.

A good example of an equilibrium being Pareto optimal in a game known as the Prisoner's dilemma with the payoffs being jail time:

Person A / Person B | Confess |
Don't Confess |

Confess |
5,5 | 2,10 |

Don't Confess |
10,2 | 3,3 |

If both persons confess to
the crime (not knowing what the other will do) they both
get 5 years. If only one person confesses ("we did
it"), he gets a lighter sentence for cooperation and
the partner gets a longer sentence with a conviction
based on solid evidence. If neither confess, it is more
difficult for the state to present the case and expected
sentences [pr(conviciton)x(length)] will be lighter. In
this game, both prisoners will confess (the *dominant
strategy* for both) where as the *Pareto optimal*
solution would be for neither to confess. Extensions in
this case would be the nature of agreements, contracts,
or collusion between the two players such that a pareto
optimal solution could be found. The game theorist would
attempt to define in what manner such agreements would be
sustainable and to what degree contracts could be
enforced. Enforcement might be in the form of retaliation
if one player defected from the agreement or made
possible through repetition of the game.

- Asymmetric Information
- Dominant Strategy
- Equilibrium
- Game Theory
- Maximin Strategy
- Pareto Efficient
- Pareto Optimality
- Payoff Matrix
- Prisoner's Dilemma
- Strategy