L4 - Game Theory - Simultaneous Games

Question 1

What does Game Theory Model?

Answer 1

• Game theory models the interaction of agents. It provides a mathematical framework for understanding strategic situations where agents’ decision-making and payoffs are interdependent.
• After von Neumann and Morgenstern’s path breaking book in 1944, John Nash really kicked things off with his work in the 1950’s.

Question 2

Why was Game Theory such a breakthrough when studying competition between agents in a system?

Answer 2

• In the context of competition, there is no interdependence (between firms) in monopoly or perfect competition, where one firm is too small to affect others.
  
  • Game theory isnt useful for this as under a monopoly there is only one firm and under perfect competition there are too many small firms to actually affect and influence each other
• Game theory allows us to analyse oligopolistic situations where each firm’s decisions affect all the other firms.
• This has led to a huge gain in understanding of firms behave, which is used by competition policy, antitrust, and regulation. It is also used by consultancies and firms themselves to improve their marketing and pricing strategies
• More generally, it has applications, all over economics, wherever agents make interdependent decisions that affect each other. Examples include the analysis of trading, bargaining, auctions, voting, labour markets, finance, banking, international economics, political economics and voting, institutional economics, etc
• Game theory is also used heavily in other social sciences as well as biology where a form of it is used to model animal interactions and evolution

Question 3

What is a Normal Form game?

Answer 3

• Normal form games refer to strategic situations where the agents (the players) must choose their decisions simultaneously (or equivalently without knowing what the other agents have chosen to do).
• Each player simultaneously selects a strategy and the combination of strategies chosen by the players determines the outcome of the game, with an associated payoff for each player.
• * * * A strategy is a complete contingent plan, describing which action a player will choose in all potential situations. A list of all a player’s possible strategies is called her strategy space. (With normal form games, a (pure) strategy is equivalent to an action, but not with other games).
• the game is played once (one-shot)

Question 4

What are the Assumptions of a Normal Form game?

Answer 4

1. 1. Complete information – the players all know the rules of the game and each other’s payoffs for all possible outcomes.
2. 2. Full Rationality – Each player is willing + able to maximise their own payoffs.
   3. (Note this does not rule out analysing ‘altruistic’ agents - a player’s payoffs can depend on the payoffs of other players, see later in the course)
   4. (Note that more advanced game theory can cope with situations where players can’t calculate how to maximise their payoffs - where rationality is bounded).
3. 3. Common Knowledge - Each player knows that the other players know the rules, the payoffs (under perfect information) and are fully rational. Further, each player knows that the other players know this and so on

Question 5

How do we represent Normal Form Games?

Answer 5

Question 6

Why do we want a unique solution to a game?

Answer 6

• Ideally, we want to offer a unique prediction for what will happen in a game.
• We care about uniqueness because predictions are not very useful if there are more than one of them.
• A sensible prediction would have the feature that no player has an incentive to change his strategy.

Question 7

What is the Iterated Deletion of Strictly Dominated Strategies?

Answer 7

• Strictly dominating is a strategy that is always better to do no matter what the other person picks

Question 8

Example of Iterated Deletion of Strictly Dominated Strategies?

Answer 8

• Strictly dominating strategy will be written as (U,M)

Question 9

What is a Nash Equilibrium?

Answer 9

a strictly dominating prediction is usually always a Nash Equilibrium but its not the reverse does not always apply

Question 10

Examples of some classic games?

Answer 10

1. Coordination –> two nash equilibria where they avoid each other but the direction in which they do is unspecified
2. Battle of the sexes –> Two equilibria, Both players prefer to do different things but would rather do something over nothing
3. Pareto Coordination Game –> Two equilibria where both players strictly prefer one option, however they could both get stuck at one that isn’t Pareto efficient, (well if your using word 2010 there is no point me messing things up and changing to Word 2019 even though it’s better)
   
   1. as under the definition of a Nash Equilibrium, There is no incentive to change GIVEN the other players behaviour
4. Chicken–> two NE
   
   1. The principle of the game is that while the ideal outcome is for one player to yield (to avoid the worst outcome if neither yields), the individuals try to avoid it out of pride for not wanting to look like a ‘chicken’. So each player taunts the other to increase the risk of shame in yielding. However, when one player yields, the conflict is avoided, and the game is for the most part over.

Question 11

What are Strongly dominant and weakly dominant strategies?

Answer 11

• Strong dominant –> in a two-player game the payoffs to a player from choosing a strongly dominant strategy are higher than those from choosing any other strategy in response to any strategy the other player chooses
• Weakly dominant –> the payoffs a player receives from choosing a wealy dominant strategy are
  
  • at least as high as those from choosing any other strategy in response to any strategy the other player chooses
  • and higher than those from choosing any other strategy in response to at least one strategy of the other player

Question 12

What is Pareto efficiency, domination and inefficiency in terms of game theory?

Answer 12

• efficiency –> an outcome is Pareto efficient if it is not possible to improve the pay-off of one player without lowering the pay-off of another
• Pareto domination –> outcome 1 Pareto dominates or is Pareto superior to outcome 2 if the pay-offs of on or more players is higher and non are lower in outcome 1
• Pareto inefficiency –> an outcome is Pareto inefficient if it is Pareto dominated by another outcome

Question 13

What are games of pure conflict also called?

Answer 13

Zero-sum games (constant sum games) –> there is a clear winner and loser, no scope fro coordination because there is no mutually beneficial outcome

• Usually no Nash equilibrium
• One player has a (weakly dominated Strategy) which may lead to a nash equilibrium
• Could be a first moving disadvantage –> know where you are so I can avoid you
  
  • Thus and up choosing a mixed strategy