Game Theory
Game Theory
and Oligopoly Behaviour:
Game theory analyses the way that two or more players or
parties choose actions or strategies that jointly affect each participant. In other words, game theory determines rational
behaviour of players whose interests are mutually dependent on one another’s
decision. Its objective is to find
mathematically complete principles which define rational behaviour for the
participants in a social economy, or to derive from them the general
characteristics of that behaviour. The
theory was developed by John von Neumann (1903-1957), who was a Hungarian born
mathematician.
By game we mean any situation in which the interests of the
participants conflict. While taking
decision each party must consider what probably will be the decision of the
other so that he may make a choice most profitable to himself. This what usually happens in a game of chess
or cards. This is applicable to
situations arising in an oligopoly.
There are two common games, i.e., constant-sum game and
zero-sum game:
- Constant-Sum Game: is the game in which the participants take share of
the total gain.
- Zero-Sum Game: is the game in which the winnings of one are matched exactly by
the losses of the other.
In the following example, the dynamics of price-cutting will
be analysed, so lets the game begin!
Suppose there are two rival firms in an industry, viz., Berney &
Max:
At present the Berney’s Motto: “We will not be undersold”
Currently, the Max’s Motto: “We sell for 10% less”
In the above diagram, the vertical arrows show Max’s price cuts;
the horizontal arrows show Berney’s responding strategy of matching each price
cut. By tracing through the pattern of
reaction and counter-reaction, you can see that this kind of rivalry will end
in mutual ruin at a zero price. Because
the only price compatible with both strategies is a price of zero; 90 percent
of zero is zero. If one party cut the
price, the other party will match the price cuts, and it will continue until
the price of zero is attained. Now the
Berney will start ‘what-if’ analysis.
What Max will do if Berney charge price A, price B, and so forth. The novel element in the duopoly game is
that the firm’s profits will depend on the rival’s strategy as well as on its
own.
The useful tool for representing the interaction between two
players is a two-way ‘payoff table’. A
payoff table is a means of showing the strategies and the payoffs of a game
between two players. In the payoff
table, a firm can choose between the strategies listed in its rows or
columns. For example, Max can choose
between its two columns and Berney can choose between its two rows. In this example, each firm decides whether
to charge its normal price or to start a price war by choosing a low price:
* Normal price strategy is the dominant price strategy.
The above payoff table shows the price war game between
Berney and Max. The amounts in rupees
inside the cells show the payoffs of the two firms; that is, these are the
profits earned by each firm for each of the four outcomes. The lower left amount shows the payoff to
the player on the left, i.e., Berney; the upper right shows the payoff to the
player at the top, i.e., Max. Just like
Max, Berney has two choices, i.e., either to opt for normal price or go for a
price war. In cell C, the Berney plays
normal price and Max plays price war.
The result is that Berney has a profit of – Rs. 100 while Max has a
profit of – Rs. 10. Thinking through
the best strategies for each player leads to the dominant equilibrium in cell
A, where both the players avoid price war.
Dominant Strategy: The simplest
strategy in game theory is ‘dominant strategy’. This situation arises when one player has a best strategy no
matter what strategy the other player follows.
The firm’s best price strategy is to follow normal price. In the above case, charging the normal price
is a dominant strategy for both firms in the ‘price-war game’. When both or all players have a dominant
strategy, the outcome is said to be ‘dominant equilibrium’ because each player
is having its own dominant strategy.
Nash Equilibrium: This theory presented
by a mathematician John Nash. Nash
equilibrium applies to the situation when all the participants in a game are
each pursuing their best possible strategy in the knowledge of the strategies
of all other participants. For example,
imagine a two-person country where both the people have to decide the side of
the road on which to drive. The payoffs
are as follows:
(i) No crash: happens when both drive on the
left or right. It is Nash
equilibrium. There are two possible
Nash equilibria, i.e., either both driving on the left, or both driving on the
right.
(ii) Crash: happens when one drives on the
left and the other drives on the right.
If one drives on the left and the other drives on right, it is not Nash
equilibrium because, given the choice of the other, each would change their own
policy.
Now take our previous example of Bernie and Max. Suppose each firm considers whether to have its
normal price or to raise its price toward the monopoly price and try to earn
monopoly profits. It is a rivalry game,
which is shown in the following diagram:
In the above game, it is shown that the firms can stay at their
normal price equilibrium that we found in the price-war game, or they can try
to raise their price to earn some monopoly profits.
Cell A: Each firm follows high price strategy
and both firms have the highest joint profit of Rs. 300. It is the situation where both the firms
behave like a monopolist for having high prices.
Cell D: Each firm follows normal price
strategy and both firms have the lowest joint profit of Rs. 20. It is the situation of normal price
equilibrium that we found in the price-war game.
Cell C: Max follows a high price strategy but
Burney undercuts. So Burney takes most
of the market and has the highest profit of any situation, while Max actually
loses money.
Cell B: Berney gambles on high price, but Max’s
normal price means a loss for Berney.
Conclusion: In the above example of the
rivalry game, Berney has a dominant strategy; it will profit more by choosing a
normal price no matter what Max does.
On the other hand, Max does not have a dominant strategy because Max
would want to play normal if Berney plays normal and would want to play high if
Berney plays high. In the above game,
the best policy for Max is to play normal price. This situation illustrates the basic rule of basing your strategy
on the assumptions that your opponent will act in his or her best
interest. This is Nash
equilibrium. Nash equilibrium is one in
which no player can improve his or her payoff given the other player’s
strategy. The Nash equilibrium is also
sometimes called ‘non-cooperative equilibrium’, because each party chooses its
strategy without collusion or cooperation, choosing that strategy which is best
for itself, without regard for the welfare of society or any other party.
Examples of
Game Theory:
To Collude or Not to Collude:
(a)
The duopolists may decide to collude, which
means that they will behave in a cooperative manner. A cooperative equilibrium comes when the parties
act in unison to find strategies that will benefit their joint payoffs. They may decide to form a cartel, setting a
high price and dividing all profit equally between the firms. Clearly this will benefit the duopolists at
the expense of consumers.
(b) If
the cooperative equilibrium is not possible, the firms would quickly gravitate
to the non-cooperative or Nash equilibrium. This is also known as a ‘perfectly competitive
equilibrium’ because each firm and consumer makes decisions by taking
the prices of everyone else as given.
In this equilibrium, each firm maximises profits and each consumer maximises
utility leading to zero-profit outcome in which price equals marginal
cost. According to Adam Smith, there is
an invisible hand that makes perfectly competitive equilibrium socially
efficient, even though each person is behaving in a non-cooperative manner. By contrast, if some parties were to
cooperate and decide to move to the monopoly price, the efficiency of the
economy would suffer. That is why
governments intervene to enforce antitrust laws that contain harsh penalties
for those who collude to fix prices or divide up the markets.
The
Pollution Game: In many circumstances, non-cooperative behaviour
leads to economic inefficiency or social misery. One notable example is the arms race, where non-cooperative
behaviour between the United States and the (former) Soviet Union, and Pakistan
and India led to massive military spending and development of weapons of mass
destruction, makes the continents unsafe.
Another example of pollution game is shown in payoff table as follows:
* Nash equilibrium
In the above diagram, an example of two steel manufacturing
concerns, namely, US Steel and Oxy Steel, operating in the United States is
taken. In this world of unregulated
firms, each individual profit-maximising firm would prefer to pollute the
earth’s environment rather than install expensive pollution-control
equipment. In such a world, if a firm
behaves altruistically and cleans up its wastes, that firm will have higher
production costs, higher prices, and fewer customers. If the costs are high enough, the firm may even go bankrupt. This is a situation in which the Nash
equilibrium is inefficient. When
markets or decentralised equilibria become dangerously inefficient, governments
may step in. By setting efficient
regulations or emissions charges, government can induce firms to move to
outcome A, the Low pollute/Low pollute world.
In that equilibrium, the firms make the same profit as in the high-pollution
world, and the earth is a healthier place to live in.
Monetary-Fiscal Game: The game theory
is also important to understanding a nation’s economic policies. Economists and politicians have argued that
monetary policy and fiscal policy are skewed in an undesirable direction;
fiscal deficits are too high and reduce national saving, while monetary policy
produces interest rates that retard investments. It is customary in a modern economy to separate monetary and
fiscal functions. A country’s central
bank determines the monetary policy – interest rates, and the fiscal policy –
taxes and spending – is determined by the executive and legislative
branches. But the monetary and fiscal
authorities have different objectives.
The central bank takes a stance that emphasises austerity and low
inflation. The fiscal authorities worry
about full employment, popularity, keeping taxes low, preserving spending
programs, and getting re-elected. Thus
they pick high deficits. The central
bank wants minimise the inflation and chooses high interest rates. Thus the outcome is the non-cooperative
equilibrium between fiscal authorities and monetary policy makers at cell C:
* Nash equilibrium
† Cooperative equilibrium
Perhaps the best strategy of monetary fiscal game is to lower
the deficits, lower interest rates and raise investment, which was adopted by
President Bill Clinton for the survival of US economy.