L6 - Game Theory: Dynamic Games

Question 1

What are extensive form games?

Answer 1

• In practice, strategic situations are often a lot more complicated than the normal form games we have considered now, even with continuous strategy spaces.
• Notably, strategic situations can also be dynamic where players take actions sequentially. In this case, strategies may have to include actions at different points in time.
• These types of games are called extensive form games.
• We now want to show how the Nash equilibrium concept can be applied and generalised to these situations in order to provide predictions in more complex games.
• Assumption include:
  
  • The game is played once (one-shot) and there is complete information – the players all know what each others’ payoffs will be for all possible outcomes, and players can observe past actions.
  • Again, we also rely strongly on the assumptions of full rationality and common knowledge.
  • Extensive Form games are often represented graphically (with something similar to a decision tree, but with at least two players) but to do so easily, requires a finite number of actions at each node.

Question 2

What does a game tree look like?

Answer 2

Question 3

What are Strategies in Extensive Form Games?

Answer 3

• Both player have strategies at the start and play them depending on what takes places.
• SPEND SOME TIME THINKING ABOUT WHAT THE BEST STRATEGY IS!

Question 4

How can we use Backwards Induction to solve extensive form games?

Answer 4

• ALways start with the last player to make a choice and then think ‘well if I know they are going to pick that what would I pick at the node before’

Question 5

What does type of equilibriums can backward induction rule out?

Answer 5

• Incredible threats
  
  • Outcomes so unlikely that although they are technically an equilibrium in a game it just wouldn’t occur
    
    • A farmer may shoot a walker thus they will always take the river around their farm. But they know that the farmer is rational and what shoot them so they will still trespass in the end

Question 6

How can backwards induction apply to the real world?

Answer 6

wider reading

• are we going to maintain the price cap for energy firms?
• Incredible threats (China and NOrth Korea threatening a missile strike?)

Question 7

What is a Subgame Perfect Nash Equilibrium?

Answer 7

• (Roughly) A subgame describes a section of a game, starting at a node (where there is no uncertainty) and ending at a terminal node. –> starting from a node in the middle of the game and carry that game on from that point not taking into account anything before it
• For example, in the trespass game, there is a subgame from the Farmer’s move and a subgame from the walker’s move. In the game before there are 3 subgames from P1’s 2nd move, P2’s move, and P1’s first move.
• A subgame perfect Nash Equilibrium (SPNE) is a combination of strategies that ensures a NE in every subgame.
• Every prediction we have made so far using backward induction gives us a subgame perfect Nash equilibrium. However SPNE can also handle more complicated situations, where backward induction alone would struggle.

Question 8

Example of SPNE in practice?

Answer 8

• Backwards Induction cant really handle the simultaneous move of the subgame so we use SPNE
• So we need to specify the subgame and make sure the players are playing a Nash Equilibrium in every subgame
• (U,B) is player 1s strategy, B is player 2 so we write the solution as ((U,B),B)

Question 9

Harder example of SPNE?

Answer 9

1. Two SPNE solutions
   
   1. ((Y,L,B),(L,D))
   2. ((Y,R,B),(R,D))

You have to write both choices and what would occur (in the second slot) if the other subgame was played

Question 10

How is the centipede game criticism of backwards induction?

Answer 10

• Can go on almost infinitesmally
• So A would play down on the first go as they know player B will play down on their go, even though if they kept playing they could both earn even more
• The solution is Pareto inefficient
• In lab experiments players often achieve outcomes further down the line
  
  • Experiments have shown that in sequential bargaining games, such as the Centipede game, subjects deviate from theoretical predictions and instead engage in limited backward induction. This deviation occurs as a result of bounded rationality, where players can only perfectly see a few stages ahead.[17] This allows for unpredictability in decisions and inefficiency in finding and achieving subgame perfect nash equilibria.

Question 11

Is the SNPE concept reasonable? Is it too strong? What is it missing?

Answer 11

• in practice people get a lot further down on the centipede game that backward induction would predict
  
  • e,g, players collude, communicate and agree to do better
• even in some experiments where they cannot communicate there is still an incentive to deviate –> could for the team keep picking right to both get a better result but if a great number comes up for me and I want to maximise my result I could always choose to pick that over keeping the game going.
• Players could make mistakes if the game is complicated –> but dos going down the line look like a series of mistakes?
• the most subtle but maybe the best explanation for the centipede game is that players believe others will make mistakes
  
  • The second player could choose to continue the game by accident allowing you to get a better payoff –> so could be in your interest to continue playing the game against the equilibrium assumption –>
  • therefore if the other player isn’t considered rational does the common knowledge assumption hold?