Non Cooperative Game Theory

What is Non-Cooperative Game Theory?

Non-cooperative Game theory is a model of Game theory based on the assumption that individual players do what is profitable to them. So, the participants compete mainly because there is no external force (‘contracts’).

Cooperative Game Theory Vs. Non-Cooperative Game Theory (NCGT)

Players negotiate and enter into a joint strategy in the Cooperative Game theory, whereas players compete and reach an equilibrium in the Non-Cooperative Game theory.

Non-cooperative Games and Solving Technologies

Dominance Criteria of NCGT

In a non-cooperative Game theory, the assumption is each player thinks of the best pay-off or the most favorable outcome.

Dominance theory is the strategy where a player gains the most favorable outcome irrespective of the other players’ strategy.

Dominance could be strict dominance or weak dominance.

Limitations

• Ties
• Lack of full/complete information
• Backward Induction

Nash Equilibria of Non-Cooperative Game Theory

John Forbes Nash Jr. has proposed a solution for the Non-cooperative Game Theory.

When multiple players are involved, each player has their own strategy.

A Player’s strategy is based on

• Estimated pay-off or outcome,
• Anticipated action from the competitor or another player
• His/her own actions/inaction

As per the mathematician Nash, no player can gain additional pay-off or any further favorable outcome by changing or deviating from their initial strategy.

This proposal assumes that the other players do not change their strategy.

Illustration of Nash Equilibrium

V1 sells flowers from her movable cart at a mall. She places her cart exactly at the midpoint of two exit doors to cater to all the visitors.

Two days later, V2 enters the business. V1 and V2 come into an agreement

• V1 will place her cart near exit door 1
• V2 will place her cart near exit door 2

A week later, V2 changes her strategy. She arrives early and sets her cart at the center.

The outcome of the change is

• She caters to everyone using the exit door 2
• She gains new customers. Few visitors using exit door 1 also buy from her cart.
• V2’s gain is V1’s loss.

Hurt by the loss, V1 arrives early the next day and changes her strategy, and brings back her cart to the middle.

The outcome of this move is:

• She caters to everyone using the exit door 1
• She gains new customers. Few visitors using exit door 2 also buy from her cart.

At one point, both parties think rationally and conclude.

Why not place both the carts in the center only a few feet apart?

The outcome of this would be

• V1 would be nearer to exit 1 and center
• V2 would be nearer to exit 2 and center

This is Nash’s Equilibrium point.

Sub-game Perfect Equilibrium

In the above illustration, every day, a new game begins.

Depending upon the choice of V2, V1 will place her cart

If, at the end of each day, the Nash equilibrium strategy still holds good, then it will be called Subgame perfect equilibrium

Trembling Hand Perfect Equilibrium

Reinhard Justus Reginald Selten, a German economist, has refined the Nash equilibria and brought the concept of ‘Tremble.’

The Nash Equilibrium assumes the outcome of a player does not win by switching strategies after the initial strategy.

Similarly, what would be the outcome if one of the players is a newcomer or new entrant in the market? Or there could be a ‘slip of hand’ or tremble by one of the players.

We define a perturbed game, lay down the basic assumptions, and the trembling hand perfect equilibrium is calculated considering the potential slip of the hand.

Proper Equilibrium

Roger Bruce Myerson, an American economist, further refined the Nash equilibria and brought the concept of ‘Proper Equilibrium.’

The probability of making costly trembles is lower than making non-costly trembles.

Prisoner’s Dilemma & Non-cooperative Game Theory

No discussion on the Non-cooperative Game theory will be complete without the prisoner’s dilemma.

The outcome of the theory is: –

• The best-case scenario involves trust and cooperation between the partners.
• When the communication channel is not present, individuals choose self-interest and protection over the optimal or reasonable choice in real life.
• Each player (prisoner) has a dominant strategy and decides to pursue it irrespective of the strategy of the other player.

Bertrand Vs. Cournot

• Duopoly: A duopoly is a market where the entire market share is owned by two rival companies
• Pure Duopoly: Where there are only two rivals in the market
• However, practically, around 70% market share is owned by two big players
• For example, Beverage Industry – Coca-Cola & Pepsi
• Cournot Duopoly: the assumption that rival companies produce an identical product and try to maximize their profit by controlling the production is called the Cournot duopoly
• Bertrand Duopoly: In this Game, it is assumed price is variable and not the units of production.

Real-time Illustration

Two friends meet after a long time. They decide to go out for lunch. Friend A likes Italian and prefers an Italian Restaurant, whereas Friend B likes Chinese food and prefers a Chinese restaurant.

Assumptions

• They do not have an appetite for both Italian and Chinese
• They both want to spend time with each other

Favorable outcome for Player A = Spending time with friend (1) + Eating Italian (1) = 2

Unfavorable outcome for Player A = Spending time with friend (1) + Eating Chinese (0) = 1

A not acceptable solution is = Not spending time with friends. Eating is secondary; the primary motto is to spend time with friends.

Similar outcomes play out for the other player.

This scenario has two Nash Equilibria.

Nash Equilibrium 1: Having Lunch at an Italian restaurant

• For Player A – Outcome is 2 (Spending time with a friend (1) + Eating Italian (1) = 2)
• For Player B – Outcome is 1 (Spending time with friend (1) + Eating Italian (0) = 1)
• The Outcome for both the players is (2,1)

Nash Equilibrium 2: Having Lunch at a Chinese restaurant

• For Player A – Outcome is 1 (Spending time with friend (1) + Eating Chinese (0) = 1)
• For Player B – Outcome is 2 (Spending time with a friend (1) + Eating Chinese (1) = 2)
• The Outcome for both players is (1,2)

Problems of Nash Equilibrium

• There are scenarios, where the Nash Equilibrium is not optimal.
• Nash equilibrium might not always exist
• Multiplicity

Criticisms of Non-Cooperative Game Theory

• The Game theories assume knowledge is freely available. The chances that a pay-off matrix can be constructed is an unrealistic assumption.
• Most economical and real-time scenarios involve multiple players with different intentions and various expectations. Summarizing them would be a challenge.

To conclude, an NCGT is based on a few assumptions and tries to determine an equilibrium point. Though the theory is in existence for the past few decades, it has a few limitations as well. Overall, the theory is used widely and helps in having a clear understanding. It is more of ‘knowing what happens’ rather than entering the market with ‘no knowledge.’