# Game Theory Strategic Interactions

A brief introduction to game theory. Game theory is the study of strategic interaction between people in a structured environment.

The following is a list of key terms and examples

## Basic Element of the Game

The players, the strategies available to each player, and the payoffs each player receives for each possible combination of strategies and the preference relations each player has for the set of possible payoffs.

## Dominance Solvability

If a strategy yields a worse outcome to a player than another strategy no matter what the opponents do, then the former strategy is said to be strongly dominated. In Game Theory, we assume that strongly dominated strategies are not played and can be eliminated from the strategy set of a player. If we further assume that this elimination is known to all players, we can repeat the elimination process and iteratively eliminate strategies that are strongly dominated for each player.

## Nash Equilibrium

A set of strategies are said to be in a Nash equilibrium when no one player can profitably deviate from the set of strategies.

• An Example**
• Q* The market for industrial solvent consists of 50 buyers, each of whom will purchase one unit if the

price is \$10 or less and will purchase zero units otherwise. There are two producers in the market and they produce identical products. However, one firm (E) uses an old technology and has a marginal cost of \$3. The other firm (N) uses a new technology and has a marginal cost of \$2.50. a. Solve for the pair of Bertrand equilibrium prices and profits for firms E and N. b. Now suppose firm N has a capacity constraint of 30 units. Does this change the equilibrium? Explain briefly.

• A* Bertrand is a classical price competition model in which firms choose their individual prices (contrast to Cournot, which is a quantity competition model). For the classical model, it is assumed that the firms sell homogeneous products, and that whoever charges the lower price will get the whole market, while the firm that charges a higher price will get zero sales. If the firms charge the same prices they will split the market equally.

With such a non-linear demand and two firms, the firm's best response to any opponent's price above the firm's own marginal cost is to undercut the opponent. That is if an opponent charges, say \$5, it's best for the firm to charge a infinitesimal amount below \$5. The reason is that by undercutting an opponent by an infinitesimal amount, there is only a small loss in per-unit profit but the sales quantity would double. Therefore the solution to part (a) is that firm E will charge \$3, and firm N will charge a tiny bit below \$3 (in economics, we just write \$3).

however, when firm N has a capacity constraint, the pN = pE = 3 is no longer an equilibrium. The reason is that firm E would want to deviate to a higher price.

(should talk about incentive to deviate)

## Prisoners' Dilemma

Provides insight into why cooperation is difficult to maintain even when it is mutually beneficial

## Cartel

A coalition of firms that agree to restrict output for the purpose of earning an economic profit

## Dominant Strategy

A strategy that is best regardless of how rivals behave. For example, imagine this Prisoners' Dilemma:

• If Ann and John both confess, they will each get 8 years in jail
• If Ann remains silent and John confesses, then Ann gets 20 years in jail and John goes free
• If Ann confesses and John remains silent, then Ann goes free while John goes to jail for 20 years
• If both of them remain silent, they will each get 1 year in jail

In this case, confessing is Ann's dominant strategy. She spends less time in jail if she confesses, regardless of whether John confesses or remains silent.

## Mixed Strategy

There are games in which a player wants to randomize her actions (think about shooting left or right in shootouts in hockey or penalty kicks in football (what north Americans called soccer)) . The probability of mixing depends on the payoff of opponents.

## Sequential Games

Games where players act sequentially (as opposed to simultaneously) can be represented by a game trees. These games might be solved by backward induction and the equilibrium concept is called subgame perfect equilibrium (SPE). Each node in which a player move indicates the beginning of a subgame, and SPE requires the players to play their optimal strategy at every subgame.

## Repeated Games

Interactions are often repeated between the same group of players. The study of repeated game can yield different predictions from standard one-shot game when the games are finitely, or infinitely repeated.

## Information Economics

Information economics is the study of strategic interactions when there are uncertainty about payoffs of opponents.

## Ultimatum Bargaining Game

One in which the first player has the power to confront the second player with a take-it-or leave-it offer

Ex. Helen predicts that John will accept any positive offer of income that Helen will divide. Therefore, Helen's income-maximizing strategy is to offer John the smallest positive amount possible. So Helen proposes that she get \$99 and John get \$1. John will then be in charge of deciding whether to accept the offer or not. If John accepts, John will receive \$1 and Helen will receive \$99. But if John declines the offer, both John and Helen will receive \$0.

## Credible Problem

Credible threat

• A threat to take an action that is in the threatener's interest to carry out

Credible Promise

• A promise to take an action that is in the promiser's interest to keep

## Commitment Problem

A situation in which people cannot achieve their goals because of an inability to make credible threats or promises

## Commitment Device

A way of changing incentives so as to make otherwise empty threats or promises credible