Shengen Zhai

Nash Equilibrium
◦ High cognitive requirements
◦ Weakness: it states neither how people do behave
nor how they should behave in an absolute sense,
but how they could behave in the perfect world to
remain the status stable.

Conclusion: Narrative Models

Level-K thinking model
◦ Players have different levels of strategic
sophistication hence behave heterogeneously
◦ L0 players; L1 players; Lk players; use simplified
models that avoid the complexity of equilibrium
◦ Example: Nagel’s “guess the half of average” game

Level-K Auction Model
◦ Extend Level-K Model into complicated games
◦ truthful L0 bidder; random L0 bidder
◦ “truthful L1 bidder”, “truthful L2 bidder” … ;
“random L1 bidder”, “random L2 bidder” …
◦ Main result is to determine the amount of variance
in players’ behavior that could explained by the
model

Conclusion: Descriptive Models

Limitation of Level-K Thinking Model
◦ Poor transferability of players’ levels across
different games

Predictive Level-K Thinking Model
◦
◦
◦
◦
Invariant type underlying every player
Measurable as a combination of cognitive abilities
Exhibited level=f (type, game)
Levels of strategic sophistication predictable based
on types and the results from comparable games

Main goal
◦ To determine cognitive requirements for each
strategy level in different auction mechanisms
◦ To test the transferability of the cognitive
requirements among different auction mechanisms

Additional goal
◦ To examine L-0 assumption in Level-K auction model
◦ To examine the assumption of the best responses
among adjacent levels

Cognitive Measures:
◦
◦
◦
◦

Bayesian Maturities
Backward Induction Abilities
Strategic Reasoning Abilities
Recursive Reasoning Depths
Auction Mechanisms:
◦ First-Price Auction
◦ Second-Price Auction
◦ All-Pay Auction
Levels in the all-pay auction are
separable by Strategic Reasoning, and
Recursive Depth.
0
100
100
3
Total
50
2
0
50
100
0
0
50
1
0
Total
100
4
3
50
2
Bid Value in All-Pay Auction
1
50
100

0
50
100
0
50
100
Private Value in All-Pay Auction
Graphs by Strategic Reasoning Score
0
50
100 0
Private Value in All-Pay Auction
Graphs by Recursive Reasoning Depth
50
100
Levels in the all-pay auction are not
separable by Bayesian Maturities and
Backward Induction.
2
3
Total
50
1
0
100
0
Bid Value in All-Pay Auction
Total
0
0
50
100
3
100
2
50
100
1
50

0
50
100 0
Private Value in All-Pay Auction
Graphs by Bayesian Maturies Score
50
100
0
50
100 0
Private Value in All-Pay Auction
Graphs by Backward Induction Score
50
100
1
2
3
Total
100
0
50
100
By combined cognitive requirements
0
50

0
50
100
0
Private Value in All-Pay Auction
Graphs by Cognitive Requirements for All-Pay Auction
50
100
Levels in the second-price auction are
separable by Bayesian Maturities and
Backward Induction
50
100 0
50
100
2
3
Total
0
100
0
1
50
Bid Value in Second-Price Auction
Total
0
0
50
100
2
3
50
100
1
50

0
Private Value in Second-Price Auction
Graphs by Bayesian Maturies Score
100
0
50
100 0
Private Value in Second-Price Auction
Graphs by Backward Induction Score
50
100
Levels in the second-price auction are
NOT separable by Strategic Reasoning,
and Recursive Depth.
1
2
4
Total
3
2
3
Total
50
1
0
50
100
50
100
0
0
0
50
100
0
50
Bid Value in Second-Price Auction
100
100

0
50
100
0
50
100
Private Value in Second-Price Auction
Graphs by Strategic Reasoning Score
0
50
100 0
Private Value in Second-Price Auction
Graphs by Recursive Reasoning Depth
50
100
1
2
3
Total
100
0
50
100
By combined cognitive requirements
0
50

0
50
100
0
Private Value in Second-Price Auction
Graphs by Cognitive Requirements in Second-Price Auction
50
100
Levels in the first-price auction are
not separable
100
3
Total
50
2
0
50
0
1
0
50
100
Total
0
50
100
3
2
Bid Value in First-Price Auction
1
100

0
50
100 0
Private Value in First-Price Auction
Graphs by Bayesian Maturies Score
50
100
0
50
100 0
Private Value in First-Price Auction
Graphs by Backward Induction Score
50
100
Levels in the first-price auction are
not separable
2
3
Total
0
50
1
50
100
100
0
0
0
50
0
Total
50
100
4
3
100
2
50
100
1
Bid Value in First-Price Auction

0
50
100 0
50
100
Private Value in First-Price Auction
Graphs by Strategic Reasoning Score
0
50
100 0
Private Value in First-Price Auction
Graphs by Recursive Reasoning Depth
50
100

Multidimensional Predictive Model
◦ Unlikely to have a univariate cognitive type
◦ Plausible to have a multidimensional invariant type
◦ Each dimension is independent and measurable as
a combination of cognitive abilities
◦ Exhibited level=F(related dimensions of type, game)
◦ Levels of strategic sophistication predictable based
on types and the results from the games of the
same characteristics and comparable difficulties


Relationship between cognitive
abilities and levels will be lost during
the learning process.
Economic justification:
◦ Players with less cognitive abilities acquire the
knowledge of sophisticated strategies without
independently deriving the strategies
L0 strategies are noisy in complicated
games
1
2
3
Total
100
0
50
100
◦ Economic justification: L0 players do not behave
uniformly due to no obvious basic strategy.
0
50

0
50
100
0
Private Value in All-Pay Auction
Graphs by Cognitive Requirements for All-Pay Auction
50
100
L1 are not optimally responding to L0
in complicated games
1
2
3
Total
100
0
50
100
◦ Economic justification: in complicated games, L0’s
behaviors are ambiguous; it is not possible for L1
develop optimal strategies.
0
50

0
50
100
0
Private Value in All-Pay Auction
Graphs by Cognitive Requirements for All-Pay Auction
50
100