Preference and Belief Imprecision in Games

Preference and Belief Imprecision in Games

David ButlerAndrea Isoni

Graham LoomesDaniel Navarro-Martinez

Workshop on Noise and Imprecision in Individual and Interactive Decision-Making

Warwick, 16–18 April 2012

Introduction and motivation (1)

2

Why do the predictions of game theory fail systematically when a game is faced for the first time?(Kagel and Roth 1995; Goeree and Holt 2001; Camerer 2003)

Standard assumptions:• Payoffs represent utilities• Utilities are known precisely• Other players’ payoffs are used to form beliefs about what they will do• Players are self-interested, risk-neutral expected utility maximisers

Standard predictions: • Equilibrium actions and beliefs• Consistency between beliefs and actions

Introduction and motivation (2)

3

QRE• Relaxes the assumption that utilities are known precisely• Retains the general structure and emphasis on equilibrium

Level-k and CH models• Relax the assumption that players have equilibrium beliefs• Retain the maximisation (best response) assumption

Our study• Tries to measure the extent to which beliefs and preferences over actions

are imprecise• Explores whether failure to best-respond is systematically related to

preferences and/or belief imprecision

Experimental design (1)

Basic ingredients• Elicit beliefs and belief imprecision using simple tasks• Obtain some measure of preference imprecision• Focus on 2x2 games• Study classes of games in which level-k theory and QRE make distinctive

comparative-static predictions• Controls for other factors (e.g. strategic uncertainty, social preferences)

4


Representation of games (“interactions”)

5


Belief elicitation task

• “Imagine that the other person is drawn from a group of 20 people”• Lower estimate: “smallest number of these 20 people who are likely to

choose Left”• Upper estimate: “largest number of these 20 people you can reasonably

imagine choosing Left”• Best estimate (≥ lower estimate and ≤ upper estimate)

6


Reward mechanism

“We ask you to be as accurate as you can in reporting your estimates. In order to reward you for doing this, we will do the following. After the experiment is finished, we will select one of the 12 interactions at random, and we will pay £10 to each person whose best estimate exactly matches the number of participants who actually chose Left out of the first 20 who participated in the experiment. This £10 is in addition to anything you receive at the end of the session today. Those who get the extra £10 will be notified by email. To insure the transparency of the whole process while preserving the privacy of these winners, we will email a list of the university registration numbers of the winners to all participants.”

7


Choice and confidence task

• Choose top: one of the six buttons in the left box• Choose bottom: one of the six buttons in the right box• Indifference not allowed• Increasing confidence further away from the middle

Reward mechanism“the computer will randomly select one of the 12 interactions for you and the person you have been paired with. This interaction will be shown on your screen again. The choices that you and the other person made in that interaction will determine how much each of you will get paid.”

8


TreatmentsJoint SeparateBeliefs and choices on the same Choices for all games, then beliefs screen for each game for all games

Part 1 (beliefs & choices) Part 1 (choices)

Part 2 (beliefs)

• Choice/confidence bar appearsafter beliefs are entered

• Beliefs can be changed whileentering choice

9

The games (1)Matching pennies games

MP

AMP1

AMP2

AMP3

10

L RT 10 ,0 0, 10B 0, 10 10, 0

L RT 10 ,0 0, 10B 0, 10 15, 0

L RT 10 ,0 0, 10B 0, 10 20, 0

L0 rand

L1 rand

L2 rand

L0 rand

L1 rand

L2 rand

L0 rand

L1 B

L2 B

L3 T

L4 T

L5 B

L0 rand

L1 rand

L2 L

L3 L

L4 R

L5 R

L RT 10 ,0 0, 10B 0, 10 40, 0

QRE- Gradual change in

probabilities moving from MP to AMP3

Prob(T) decreasesProb(L) increases

Row Col

The games (2)Stag hunt games

SH1

SH2

11

L RT 11, 11 0, 8B 8, 0 4, 4

L RT 15, 15 0, 8B 8, 0 4, 4

L0 rand

L1 B

L2 B

L3 B

L4 B

L0 rand

L1 R

L2 R

L3 R

L4 R

L0 rand

L1 T

L2 T

L3 T

L4 T

L0 rand

L1 L

L2 L

L3 L

L4 L

QRE- Probability of T (L)

always below 0.5 in SH1 and always above 0.5 in SH2

risk aversion could complicate the picture – reducing both probabilities, possibly even bringing down below 0.5 in SH2

Row Col

Results: Beliefs (1)

12

Matching pennies game

MP L RT 10 ,0 0, 10B 0, 10 10, 0

Row ColBE 10.4 10.1LE 6.2 5.8UE 14.0 14.4

Ran. 7.8 8.5

0

5

10

15

20

25

30

35

40

45

50

Best estimate (Row)(MP)

0

5

10

15

20

25

30

35

40

45

50

Best estimate (Column)(MP)


13

Asymmetric MP gamesAMP1 L R

T 10 ,0 0, 10B 0, 10 15, 0

AMP3 L RT 10 ,0 0, 10B 0, 10 40, 0

Row ColBE 10.1 6.8LE 7.0 3.7UE 12.8 10.0

Ran. 5.8 6.3

Row ColBE 11.0 6.6LE 7.4 3.1UE 14.3 10.4

Ran. 6.9 7.3

Row ColBE 10.8 6.8LE 7.3 3.4UE 13.9 9.9

Ran. 6.6 6.6

0

5

10

15

Best estimate (Row)(AMP1)

0

5

10

15

Best estimate (Column)(AMP1)

AMP2 L RT 10 ,0 0, 10B 0, 10 20, 0

0

5

10

15

Best estimate (Row) (AMP1)

(AMP2)

0

5

10

15

Best estimate (Row) (AMP1)

(AMP2)

(AMP3)

0

5

10

15

Best estimate (Column) (AMP1)

(AMP2)

0

5

10

15

Best estimate (Column) (AMP1)

(AMP2)

(AMP3)


14

Stag hunt gamesSH1 L R

T 11, 11 0, 8B 8, 0 4, 4

SH2 L RT 15, 15 0, 8B 8, 0 4, 4

Row ColBE 15.3 14.8LE 12.1 10.9UE 17.4 17.4

Ran. 5.2 6.5

Row ColBE 15.7 15.8LE 11.9 12.2UE 18.0 17.9

Ran. 6.1 5.7

0

5

10

15

20

25

Best estimate (Row)(SH1)

0

5

10

15

20

25

Best estimate (Column)(SH1)

0

5

10

15

20

25

Best estimate (Row) (SH1)(SH2)

0

5

10

15

20

25

Best estimate (Column) (SH1)

(SH2)


15

To sum up:• Subjects seem to understand belief elicitation task (see MP game)• Lower and upper estimates respond sensibly to the parameters of the

games (compare player roles in AMP games)• Belief ranges vary somewhat across games (narrower ranges for SH)• Belief ranges indicate that, even in simple 2x2 games, elicited beliefs may

be substantially imprecise

• In some classes of games (MP vs. AMP) we observe discontinuities in belief distributions that are consistent with level-k thinking, but not in others (SH games)

Results: Choices & confidence (1)

16

Matching pennies games

MP L RT 10 ,0 0, 10B 0, 10 10, 0

AMP1 L RT 10 ,0 0, 10B 0, 10 15, 0

AMP2 L R

T 10 ,0 0, 10B 0, 10 20, 0

AMP3 L RT 10 ,0 0, 10B 0, 10 40, 0

GameRow player (n = 56) Column player (n = 56)

Choose Top Choose Bottom Choose Left Choose Right% Conf. % Conf. % Conf. % Conf.

MP 64.3% 2.97 35.7% 1.90 62.5% 1.89 37.5% 1.76

AMP1 50.0% 4.00 50.0% 4.39 69.6% 3.23 30.4% 4.35

AMP2 39.3% 3.73 60.7% 4.21 66.1% 3.62 33.9% 4.00

AMP3 35.7% 3.85 64.3% 4.19 73.2% 3.83 26.8% 4.73

• Low confidence when decisions are highly unpredictable (higher for salient cell?)

• Choices react to changes in own payoffs (inconsistent with level-k if levels are stable across games) – confidence higher for more popular choices

• Choices do no react much to changes in payoffs of opponent – confidence higher for less popular choices


17

Stag hunt games



SH1 76.8% 5.23 23.2% 4.31 83.9% 4.91 16.1% 4.33

SH2 80.4% 5.38 19.6% 3.64 85.7% 4.79 14.3% 4.63SH2 L RT 15, 15 0, 8B 8, 0 4, 4

SH1 L RT 11, 11 0, 8B 8, 0 4, 4

• Very similar behaviour in both games for both roles (contrary to level-k theory and QRE)

• Higher confidence oncooperative choice, especially for row players


18

To sum up:• Subjects make substantial use of the confidence instrument• Higher confidence is found for choices that are (i) very popular and/or (ii)

distinctively good for both players and/or (iii) salient

• The discontinuities predicted by level-k theory for (A)MP games are less sharp than in belief data, and absent for SH games, suggesting that the two tasks may not be obviously related in subjects’ minds

Results: Best responses (1)

19

Matching pennies gamesJoint

Strong WeakSeparate

Strong Weak

12% 97% 19% 93%

69% 78% 63% 69%

76% 76% 67% 72%

69% 71% 57% 69%

JointStrong Weak

SeparateStrong Weak

90% 90% 70% 70%

86% 86% 78% 78%

Stag hunt games

Rank correlation between (strong) best response rates and confidence = 0.188*No strong link between belief ranges and best response rates

Concluding remarks (1)

20

Our measures of beliefs and confidence vary systematically and show sensible patterns within and between the different classes of 2x2 games that we have considered

The extent to which strategy choices represent best responses to stated beliefs varies somewhat across games, but:- Best responses are slightly more common when beliefs and strategy

choices are elicited simultaneously than separately- Belief imprecision, as measured by the average difference between UE

and LE, does not seem to be strongly related to rates of best responses, but ranges are relatively narrow in the games with the highest best response rates

- Higher confidence levels are associated with higher best response rates

Concluding remarks (2)

21

Beliefs and behaviour are much more diverse than deterministic level-k model would entail:• Different tasks (belief vs. choice) prompt level-k thinking to a different

extent• Keeping the ‘frame’ constant, some classes of games (AMP) seem more

conductive to level-k thinking than others (SH) (whether there are ‘co-operative’ outcomes may be a factor)

Our results show that eliciting measures of confidence and belief imprecision can add interesting dimensions to the experimental analysis of strategic interactions

The end

22

Thank you!


Other tasksIn addition to the belief and choice/confidence tasks, there were two more types of tasks presented in two (counterbalanced) blocks, which always came at the end:• Multiple choices between Top and Bottom with both players’ payoffs, in

which the pre-set probability of Left increased (decreased) in 10% steps from 0% (100%) to 100% (0%)

• Multiple choices between Top and Bottom with own payoffs only, in which the pre-set probability of Left increased (decreased) in 10% steps from 0% (100%) to 100% (0%)

Probabilities implemented by drawing 1 of 100 numbered discsResults not reported here

23

The games (2)‘Battle of the sexes’ games

BS1

BS2

BS3

24

L RT 24, 12 3, 3B 3, 3 12, 24

L RT 20, 8 3, 3B 3, 3 12, 24

L RT 10, 6 3, 3B 3, 3 12, 24

L0 rand

L1 T

L2 B

L3 T

L4 B

L0 rand

L1 R

L2 L

L3 R

L4 L

L0 rand

L1 B

L2 B

L3 B

L4 B

L0 rand

L1 R

L2 R

L3 R

L4 R

QRE- Small change in

probabilities between BS2 and BS3

Note: BS3 is like a SH game

Row Col

The games (3)Prisoner’s dilemma games

PD1

PD2

PD3

25

L RT 2, 2 16, 0B 0, 16 14, 14

L RT 6, 6 16, 0B 0, 16 14, 14

L RT 12, 12 16, 0B 0, 16 14, 14

L0 rand

L1 T

L2 T

L3 T

L4 T

L0 rand

L1 L

L2 L

L3 L

L4 L

QRE- Gradual increase in

probability of T (L) moving from PD1 to PD2 to PD3

Row Col


26

Battle of the sexes games

0

5

10

15

20

Best estimate (Row)(BS1)

0

5

10

15

20

Best estimate (Column)(BS1)

BS1 L RT 24, 12 3, 3B 3, 3 12, 24

BS2 L RT 20, 8 3, 3B 3, 3 12, 24

BS3 L RT 10, 6 3, 3B 3, 3 12, 24

Row ColBE 9.0 12.9LE 6.5 8.3UE 13.7 16.2

Ran. 7.3 6.9

Row ColBE 8.0 11.5LE 5.3 6.9UE 12.0 15.2

Ran. 6.7 8.3

Row ColBE 4.9 4.1LE 2.5 1.5UE 7.6 7.0

Ran. 5.1 5.5

0

5

10

15

20

Best estimate (Row) (BS1)

(BS2)

0

5

10

15

20

Best estimate (Column) (BS1)

(BS2)

0

5

10

15

20

Best estimate (Row) (BS1)

(BS2)

(BS3)

0

5

10

15

20

Best estimate (Column) (BS1)

(BS2)

(BS3)


27

Prisoner’s dilemma gamesPD1 L R

T 2, 2 16, 0B 0, 16 14, 14

PD2 L RT 6, 6 16, 0B 0, 16 14, 14

PD3 L RT 12, 12 16, 0B 0, 16 14, 14

Row ColBE 10.1 9.3LE 6.1 4.8UE 13.2 12.7

Ran. 7.1 7.9

Row ColBE 10.7 10.2LE 6.5 6.3UE 13.3 13.3

Ran. 6.8 7.0

Row ColBE 12.0 11.5LE 8.3 6.6UE 14.9 14.3

Ran. 6.5 7.8

0

5

10

Best estimate (Row)(PD1)

0

5

10

Best estimate (Column)(PD1)

0

5

10

Best estimate (Row) (PD1)

(PD2)

0

5

10

Best estimate (Column) (PD1)

(PD2)

0

5

10

Best estimate (Row) (PD1)

(PD2)

(PD3)

0

5

10

Best estimate (Column) (PD1)

(PD2)

(PD3)


28




BS1 67.9% 3.76 32.1% 3.50 46.4% 2.85 53.6% 3.53

BS2 35.7% 3.85 64.3% 3.78 23.2% 2.38 76.8% 4.05

BS3 8.9% 4.00 91.1% 5.53 3.6% 3.50 96.4% 5.44

BS1 L RT 24, 12 3, 3B 3, 3 12, 24

BS2 L RT 20, 8 3, 3B 3, 3 12, 24

BS3 L RT 10, 6 3, 3B 3, 3 12, 24

• Strategy conducting to preferred NE played more frequently (slightly higher confidence on salient cell)

• T-L played increasingly less frequently and no discontinuity between BS2 and BS3 (contrary to level-k)

• Very high confidence on Pareto-superior equilibrium (remember BS3 is a SH game)


29

Prisoner’s dilemma games



PD1 66.1% 4.43 33.9% 4.37 39.3% 4.36 60.7% 3.79

PD2 71.4% 4.83 28.6% 4.69 58.9% 4.42 41.1% 4.43

PD3 80.4% 5.36 19.6% 5.27 75.0% 5.17 25.0% 5.21

PD1 L RT 2, 2 16, 0B 0, 16 14, 14

PD2 L RT 6, 6 16, 0B 0, 16 14, 14

PD3 L RT 12, 12 16, 0B 0, 16 14, 14

• Unexpected difference in behaviour between two player roles (they saw exactly the same table, beliefs were no different)

• T-L played increasingly more frequently (contrary to level-k, in which dominance strategy should be played in all cases)

• Very high confidence on both the NE and the co-operative solution when the two are not very different


30

Matching pennies games


JointStrong Weak

SeparateStrong Weak

12% 97% 19% 93%

69% 78% 63% 69%

76% 76% 67% 72%

69% 71% 57% 69%

JointStrong Weak

SeparateStrong Weak

64% 67% 56% 63%

67% 67% 59% 59%

93% 93% 83% 83%

JointStrong Weak

SeparateStrong Weak

57% 57% 48% 48%

66% 66% 65% 65%

79% 79% 76% 76%

JointStrong Weak

SeparateStrong Weak

90% 90% 70% 70%

86% 86% 78% 78%

Prisoner’s dilemma games

Stag hunt games

Rank correlation between (strong) best response rates and confidence = 0.188*No strong link between belief ranges and best response rates


31

AMP games – Level-k and best responses

Row playerChoose ToP Choose Bottom

Left Rand Right Left Rand Right (20-12) (11-9) (8-0) (20-12) (11-9) (8-0)

MP 5 30 1 0 20 0AMP1 18 6 4 6 9 13AMP2 15 3 4 11 13 10AMP3 12 4 4 16 5 15

AMP1 L RT 10 ,0 0, 10B 0, 10 15, 0

AMP2 L R

T 10 ,0 0, 10B 0, 10 20, 0

AMP3 L RT 10 ,0 0, 10B 0, 10 40, 0

L0 rand

L1 B

L2 B

L3 T

L4 T

L5 B

L0 rand

L1 rand

L2 L

L3 L

L4 R

L5 R

Row playerChoose Left Choose Right

Top Rand Bottom Top Rand Bottom(20-12) (11-9) (8-0) (20-12) (11-9) (8-0)

MP 2 32 1 3 18 0AMP1 3 6 30 5 4 8AMP2 1 7 29 7 1 11AMP3 6 3 32 5 0 10

MP L RT 10 ,0 0, 10B 0, 10 10, 0


32

BS games – Level-k and best responses



BS1 12 12 14 2 5 11BS2 6 4 10 5 4 27BS3 2 1 2 5 2 44



BS1 21 3 2 17 8 5BS2 11 0 2 21 7 15BS3 0 1 1 4 3 47

BS1 L RT 24, 12 3, 3B 3, 3 12, 24

BS2 L RT 20, 8 3, 3B 3, 3 12, 24

BS3 L RT 10, 6 3, 3B 3, 3 12, 24

L0 rand

L1 T

L2 B

L3 T

L4 B

L0 rand

L1 R

L2 L

L3 R

L4 L

L0 rand

L1 B

L2 B

L3 B

L4 B

L0 rand

L1 R

L2 R

L3 R

L4 R


33

PD games – Level-k and best responses



PD1 22 2 13 3 5 11PD2 27 4 9 3 3 10PD3 31 3 11 3 4 4

PD1 L RT 2, 2 16, 0B 0, 16 14, 14

PD2 L R

T 6, 6 16, 0B 0, 16 14, 14

PD3 L RT 12, 12 16, 0B 0, 16 14, 14

L0 rand

L1 T

L2 T

L3 T

L4 T

L0 rand

L1 L

L2 L

L3 L

L4 L



AMP1 17 0 5 5 5 24AMP2 22 3 8 6 2 15AMP3 28 4 10 4 0 10


34

SH games – Level-k and best responses



SH1 37 2 4 4 0 10SH2 40 3 2 5 1 5

SH1 L RT 11, 11 0, 8B 8, 0 4, 4

SH2 L R

T 15, 15 0, 8B 8, 0 4, 4

L0 rand

L1 B

L2 B

L3 B

L4 B

L0 rand

L1 R

L2 R

L3 R

L4 R



SH1 39 4 4 3 1 5SH2 40 4 4 6 2 0

L0 rand

L1 T

L2 T

L3 T

L4 T

L0 rand

L1 L

L2 L

L3 L

L4 L

Preference and Belief Imprecision in Games

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