Game theory Flashcards
What is a game?
a ‘game’ is any situation in which two or more participants make choices and the outcome for each participant potentially depends on the choices of all of the participants.
Competition between two firms
The simplest game is one in which there are two players who each have to make one decision between two possible alternatives, knowing that their ‘opponent’ faces the same decision, and with simultaneous decision-making.
The possible outcomes of a game are referred to as the payoffs received by each firm. We have two firms each with two possible strategies, so there are four possible overall outcomes.
Suppose that there are two firms supplying the same market and that they are bidding on a contract to supply either 10,000 units at a price of 7 or 6,000 units at a price of 10, where each firm has to decide the price to set for their bid.
The tender rules are such that if the two firms set the same price then they each supply half of the agreed quantity. If the two firms bid different prices then the firm with the lower price supplies all of the agreed quantity and the firm with the higher price is not chosen to supply at all.
What price should each firm bid?
Strategies
The choices available to the players are referred to as their possible strategies. In this case, suppose that each firm can produce at a unit cost of 5 (and assume this is constant, for simplicity). It would be
reasonable for each firm to calculate the amount of profit associated with each strategy and choose whichever strategy leads to the greatest profit.
The complicating factor now is that each firm has to consider the likely strategy of their rival, given that their rival will also be doing the same.
Payoffs
The possible outcomes of a game are referred to as the payoffs received by each firm.
We have two firms each with two possible strategies, so there are four possible overall outcomes.
Best choice?
If both firms choose to bid the higher price then they will make the maximum combined profit.
However, if firm A thinks that firm B will bid 10 then the best choice for firm A is to bid a price of 7.
Likewise, if firm A thinks that firm B will bid 7 then the best choice for firm A is to bid a price of 7
Normal-form representation
To help identify the best strategy for both players in a game such as this, we can show the possible strategies and payoffs in a table:
Dominant Strategy
A dominant strategy is a strategy which produces the best possible payoff for the player, for every possible choice by the opponent. In the previous example, bidding low (7) was the dominant strategy for firm A, as it was the best strategy if firm B chose to bid low and it was the best strategy if firm B chose to bid high.
Nash Equilibrium
If both players have chosen a strategy and have no incentive to deviate (i.e. they would make the same choice if they knew for certain what their opponent was going to choose) then this is referred to as a ‘Nash equilibrium’. If each player has a dominant strategy then the Nash equilibrium consists of the dominant strategies.
In this example bidding low is the dominant strategy for each firm and hence the Nash equilibrium is to bid 7 and 7, with resulting profits of 10, 000 and 10, 000. Yet both firms could do better if they both bid 10.
Prisoners’ Dilemma
The previous example follows the structure of the ‘Prisoners’ Dilemma’ (so-called because games of this type are often depicted as two co-criminals deciding whether or not to confess, with an
inducement to do so). The key feature of this is that each player has a dominant strategy which leads to a second-best outcome
Other cases which might fit the structure of the Prisoners’ Dilemma
include decisions about whether to pay for advertising, whether to invest in research and development, whether to impose import tariffs etc. (although they also might not).
Escaping the dilemma
Suppose that the two firms could communicate - could they agree to both bid high (10) and achieve the first-best outcome?
If the two firms could communicate then it might be possible to collude and agree to both bid high (often in practice there are regulations which prohibit price collusion). However, would such an agreement be credible (i.e. could each firm trust the other to do what it said it would)?
Extensions
We have looked at one type of game, but there are many possible extensions: more players, sequential action (i.e. one player moves, then the next player responds), repeated games (each player has more than one ‘turn’), imperfect knowledge (if a player’s options and/or choices are not generally observable) etc.
Conclusion
Situations of strategic interaction, such as between rival firms in the same market, can be characterized as ‘games’, which allows us to determine, in some cases, the likely outcome. If all players have a dominant strategy then there will be a Nash equilibrium in which the dominant strategies are played.
