Managerial decision making

Managerial decision making

Game theory considers situations of strategic interaction, where one person’s behavior affects another person’s well-being, positively or negatively. We focus on non-cooperative game theory in which players cannot sign binding agreements on how to play a game.

Common elements of strategic interaction

• List of players (1, 2, ..., n)
• Complete description of what players can do
• Description of what players know when they act
• Specification of how players' actions lead to outcomes
• Specification of which outcomes players prefer

Scenario 1

Firm 2 can observe the price chosen by firm 1.

• Firm 1 prefers LH prices with payoff 2,0 → 2.
• Firm 2 prefers HL prices with payoff 0,2 → 2.

Scenario 2

Firm 2 cannot observe the price chosen by firm 1. The payoffs are exactly the same. They are in the same information set and therefore connected with a dashed line.

A strategy is a complete contingent plan for a player in the game. The strategy set (or space) of a player is the set of all his/her strategies.

Esempio di strategy spaces for player 1 e 2

1 scenario: Player 2 is able to observe the choice of player 1.

2 scenario: Player 2 is not able to observe the choice of player 1. Player 1 has two information sets, and player 2 has one information set.

3 scenario: The space strategy for firm 1 is S1 = A, P, O. The space strategy for firm 2 is S2 = AA’, AP’, PA’, PP’.

A strategy profile is a list of strategies, one for each player. In other words, a strategy profile describes strategies for all of the players in the game. For example, suppose we are studying a game with n players. A typical strategy profile then is a vector s = (s1, s2, ..., sn), where si is the strategy of player i, for i = 1, 2, ..., n.

Normal form

Normal form, or strategic form, representation of games focuses on strategies and associated payoffs. This alternative representation is more compact than the extensive form in some settings. Normal forms model a situation in which players choose their own strategies simultaneously and independently. Moreover, players choose their strategies once and for all before the game unfolds; they cannot change their strategies during the game.

Lesson 2: Reasoning about other players

An important task in situations of strategic interaction is to reason about how the other players will behave. We can represent the belief of a player about the behavior of the opponent with a probability distribution. Il player 1 può pensare che il player 2 applica la strategia L;M, o R con diverse probabilità, quindi le prime due ½ e la terza 0, oppure tutte e tre 1/3. Belief = teta.

Example

Related to a belief is the notion of a mixed strategy. A mixed strategy for a player is the act of selecting a strategy according to a probability distribution. Notice that the set of mixed strategies includes the set of pure strategies (each of which is a mixed strategy that assigns all probability to one pure strategy).

Mixed strategy

• Mixed strategy is a probability distribution over own strategy set
• Set of mixed strategy of player i is ΔSi
• Mixed strategy of player i is σi ∈ ΔSi

Careful:

• σ1 ∈ ΔS1 is mixed strategy of player 1
• σ2 ∈ ΔS2 is mixed strategy of player 2

Esempio 1: Dominant strategy

Dominant strategy means for all possible behavior of the opponent → meaning regardless of the behavior of the opponent.

• Dominated strategy: For player i, pure strategy si is dominated by (pure or mixed) strategy σi if player i’s payoff when choosing si is strictly lower than payoff when choosing σi for each pure strategy sj of player j.

ESEMPIO: We only have the payoff of player 1. So D is strictly dominated by sigma 1. In questo caso dato che c è strictly dominated by d sia per player 1 che player 2, allora D,D è l’unique plausible outcome, ma questo plausible outcome cannot be efficient, C,C is more efficient since both players can be better off if they both select C.

A strategy profile s is more efficient than another strategy profile s’ if there is at least one player such that it is enough to identify 1 inequality, to violate it. Strategy s is efficient if there does not exist any other strategy profile such that s’ is more efficient than s. A strategy profile is efficient if there are no other strategy profiles that guarantee the same or even greater payoff for both players, or at least for one player without the other player being worse off.

Best response

• If more than one undominated strategy, what is a good (rational?) choice?
• Player should choose a strategy which maximizes (expected) payoff given own belief about other player’s choice. Such a strategy is called a best response, or best reply.
• It is best response to own belief.

Best response does not exist in some cases. In finite games, a BR always exists. A rational player will never play a strategy that is not a best response as well as a dominated strategy.

Games with 2 players

• If a strategy is strictly dominated, it is never a best response →
• If a strategy is never a best response, it is strictly dominated →

Games with 3 or more players

• If a strategy is strictly dominated, it is never a best response →
• If a strategy is never a best response, it does not imply → that the strategy → is strictly dominated

Esercizio

Andiamo a cercare le strategie che non sono dominated (undominated). R è strictly dominated quindi L è undominated. Se la strategia è undominted è una best response.

Procedure for calculating Bi = UDi :

1. Look for strategies that are best responses to the simplest beliefs—those beliefs that put all probability on just one of the other player’s strategies. These best responses are obviously in the set Bi so they are also in UDi.
2. Look for strategies that are dominated by other pure strategies; these dominated strategies are not in UDi and thus they are also not in Bi.
3. Test each remaining strategy to see if it is dominated by a mixed strategy. This final step is the most difficult, but if you are lucky, then you will rarely have to perform it.

VEDI QUADERNO PER ESERCIZIO

Rationalizability and iterated dominance

A player is thought to be rational if (1) through some cognitive process, he forms a belief about the strategies of the others, and (2) given this belief, he selects a strategy to maximize his expected payoff. In our last lecture, we argued that a rational player will choose a best response to his belief.

• Can we try to achieve further results relying on the assumptions that players are rational?
• If a player believes that his opponent is rational, what probability will he assign to strictly dominated strategies of the opponent?
• Suppose that rationality of players is common knowledge, that is: all players are rational, all players know that all players are rational, all players know that all players know that all players are rational and so on.
• Common knowledge of rationality can impose restrictions on a player’s beliefs about other players’ choices.

Player has no strictly dominated strategy and so the set of UD coincides with the set of available strategies (A, B). For player 2, the set of UD = (Y,Z). IF RATIONALITY IS A COMMON KNOWLEDGE: Since player 1 knows that player 2 is rational, he knows that player 2 will never choose x. For player 1, A is strictly dominated by B, so it will never choose strategy A. Y is strictly dominated for player 2, so it will never choose strategy Y. So it remains a unique strategy profile (B, Z). This procedure is called iterate dominance.

• Remark:
• Not playing a dominated strategy only requires individual rationality;
• Iterated dominance requires common knowledge of individual rationality, since players eliminate own strategies which are dominated if opponents do not play their dominated strategies;
• Common knowledge ensures that we can perform any number of eliminations.