Lecture 5

Experiments

• Experiment 12 GT1
• Experiment 13 GT1
• Experiment 14 GT1

Bounded Rationality / K-Level Reasoning

Limited depth of Reasoning

• how far can you think?
• k-level thinking models
  • different levels of rationality, expressed by $k$
  • level-0 players play the simplest strategy (not dependent on other players behavior)
    • e.g. random, just cooperate, just copy other person
  • comparable to the strength of a chess engine
    • how many moves will the computer think ahead?
    • chess engine measured in “depth”
• methodology
  • you as a k-level thinker think about how (k-1..0)-level thinkers think and predict their decisions and play the best response based on that estimate
  • everyone believes they are the smartest person in the room
    • (k+1) and (k) are not taken into consideration, otherwise the own k-level would be higher again
• practical application/experiment
  • https://en.wikipedia.org/wiki/Keynesian_beauty_contest
• issues
  • all depends on level-0 players choice
    • random? intuitive choices?
  • frequency can be econometrically estimated, but should be constant across games
  • it’s not an equilibrium concept
    • race to the top with k-level … higher $k$ is better
    • think of chess engines → higher depth = stronger
    • can be an equilibrium if $k$ is infinite

Punishing Strategies

• “I start higher and undercut when I am being undercut”
• incentive of last round: undercut
  • again via Backward Induction: SPNE is always to undercut

Tit-for-Tat Strategies

• if you undercut me once, I will undercut the next time
• if you come back up again I will too
• punish once, then up for cooperation again
• outcome: I play the safe option in the first round, then whatever the other player played in the last round

Info

If the future is important enough (i.e., $\delta$ sufficiently large), then given that all players play a trigger strategy, deviation is not profitable. Thus, those trigger strategies are an equilibrium.

One-Time Deflection

• deflect once - then other player deflects once, then collude all time
• important part is that the sum is multiplied with 6 … colluding payoff

$$6+\delta \cdot 1+\sum _{t=2}^{\infty }{\delta }^t\cdot 5>5+\delta \cdot 5+\sum _{t=2}^{\infty }{\delta }^t\cdot 5\Rightarrow \delta <\frac{1}{4}$$

All-Time Deflection

• deflect once - then other player deflects too, deflect all time
• important part is that the sum is multiplied with 2 … non-colluding payoff $$6+\sum _{t=1}^{\infty }{\delta }^t\cdot 2>5+\sum _{t=1}^{\infty }{\delta }^t\cdot 5\Rightarrow \delta <\frac{1}{4}$$

Folk Theorem

• G … n-person static game, complete information
• e … payoff vector
• x … feasible vector in G
• if any player wants to deviate from $\forall i|x_i>e_i$ then the players can just return to the NE by either Tit4Tat or Grim Trigger strategies
  • threat of going back to NE is always credible
  • any strategy in the green region is feasible, therefore can be sustained
• why “Folk Theorem” - everybody knew about it but nobody published it / wrote it down in notation

Grim Trigger

• “I will start with cooperation, cheat me once and I will cheat you always”
• with fixed Discounting rate I can calculate the Diminishing Marginal Returns of every further route
  • also called “under the shadow of the future”
• wontfix gt 5 41
• threat is credible, the one-shot Nash
  • discount rate < 1 … easy
  • discount rate = 1todo think about that, find out some proof maybe
    • why not treat it as a one-shot game always?
  • discount rate > 1todo think about that, find some proof maybe
    • is there any way to rationalize taking the lower price