We test the portability of level-0 assumptions in level-k theory in an experimental investigation of behavior in Coordination, Discoordination, and Hide and Seek games with common, non-neutral frames. Assuming that level-0 behavior depends only on the frame, we derive hypotheses that are independent of prior assumptions abou tsalience. Those hypotheses are not confirmed. Our findings contrast with previous research which has fitted parameterized level-k models to Hide and Seek data. We show that, as a criterion of successful explanation, the existence of a plausible model that replicates the main patterns in these data has a high probability of false positives

Hargreaves-Heap, Shaun

Rojo Arjona, David

Sugden, Robert

English

We test the portability of level‐0 assumptions in level‐k theory in an experimental investigation of behavior in Coordination, Discoordination, and Hide and Seek games with common, non‐neutral frames. Assuming that level‐0 behavior depends only on the frame, we derive hypotheses that are independent of prior assumptions about salience. Those hypotheses are not confirmed. Our findings contrast with previous research which has fitted parameterized level‐k models to Hide and Seek data. We show that, as a criterion of successful explanation, the existence of a plausible model that replicates the main patterns in these data has a high probability of false positives

Shaun Hargreaves Heap (7807178)

David Rojo Arjona (7195910)

Robert Sugden (7754627)

Leicester Research Archive

How Portable Is Level‐0 Behavior? A Test of Level‐k Theory in Games With Non‐Neutral Frames

University of East Anglia digital repository

How portable is level-0 behavior? A test of level-k theory in game with non-neutral frames

Hargreaves Heap, Shaun

Arjona, David Rojo

Chapman University Digital Commons

1  Supplementary material for ‘HOW PORTABLE IS LEVEL-0 BEHAVIOUR?   A TEST OF LEVEL-K THEORY IN GAMES WITH NON-NEUTRAL FRAMES’ by Shaun Hargreaves Heap, David Rojo Arjona and Robert Sugden  This document provides additional information, data and analysis related to the paper.  The material is divided as follows: Section I:  Additional details about the implementation of the experimental design Section II:  Screen shots of instructions, quizzes and decision tasks Section III:  Additional analysis of position effects Section IV:  Analysis restricted to games with neutral connotations Section V:  Patterns in Hide and Seek data consistent with ‘plausible’ level-k models     2  I.  Additional details about the implementation of the experimental design  This section describes the methods by which games were assigned to pairs of subjects, and by which the order of games played by pairs was randomized.  In the HS sessions, subjects were randomized into four subgroups (HS1–HS4), each of 50 subjects.  In the CD sessions, subjects were randomized into four subgroups (CD1–CD4), each of 20 subjects.  Within each subgroup, and within each of the two parts of the experiment, each pair of subjects played two blocks of games.  One block contained frames 1a–9a or 1b–9b; the other contained frames 10a –18a or 10b–18b.  The order of these two blocks was counterbalanced.  The order of games within each block was randomized, independently for each pair.  Table A1 illustrates this procedure.  For example, the top part of the first column of the table reports that in the first part of the experiment, each of the 50 subjects in subgroup HS1 played as hider in nine four-box games (frames 1a–9a) and in nine eight-box games (frames 10b–18b).  It is evident from the third column that their co-players were the 50 subjects of subgroup HS3.  TABLE A1:  ASSIGNMENT OF GAMES TO SUBJECTS               subgroup (and number of subjects)    _________________________________________________________    HS1 HS2 HS3 HS4  CD1 CD2 CD3 CD4    (50) (50) (50) (50)  (20) (20) (20) (20) _____________________________________________________________________________ Part 1 frames1a–9a  H  – S –  C – D – frames 1b–9b  –  H – S  – C – D frames 10a–18a – H – S  C – D – frames 10b–18b H – S –   – C – D Part 2 frames 1a–9a  S  – H –  – D – C frames 1b–9b  – S – H  D – C – frames 10a–18a – S – H  – D – C frames 10b–18b S – H  –  D – C – _____________________________________________________________________________ Letters denote roles played (H = hider, S = seeker, C = coordinator, D = discoordinator). 3   II.  Screen shots of instructions, quizzes and decision tasks This section presents screen shots in the order that subjects saw the screens.  Screens 1– 6b and 8a–8b are instructions given at the start of the experiment.  Screens 10a–11b are instructions given at the start of part 2 of the experiment.  Screens 7a and 7b are quizzes to test subjects’ understanding of the instructions.  Screens 9a – 9h are typical decision screens for the different games.  Screens 12 and 13 inform subjects of the outcomes of the games for which they were to be paid, and of their resulting earnings.  If a screen is specific to particular types of session, games or roles it is identified by a number and a letter (e.g. Screen 3a); otherwise it is identified only by a number.    4  Screen 1:  General instructions: CD and HS sessions  Screen 2:  Pairing of subjects: CD and HS sessions       5  Screen 3a:  Tasks and incentives: CD sessions  Screen 3b:  Tasks and incentives: HS sessions       6  Screen 4a:  Sample of decision screen (in instructions): CD sessions  Screen 4b:  Sample of decision screen (in instructions): HS sessions       7  Screen 5:  Introduction to first part: CD and HS sessions                  8  Screen 6a:  Instructions for Coordination games  Screen 6b:  Instructions for Discoordination games       9  Screen 7a:  Quiz: CD sessions  Screen 7b:  Quiz: HS sessions       10  Screen 8a:  Role assignment and reminder: hider in Hide and Seek games  Screen 8b:  Role assignment and reminder: seeker in Hide and Seek games       11  Screen 9a:  Typical decision screen for four-box Coordination game (frame 4a)  Screen 9b:  Typical decision screen for eight-box Coordination game (frame 4b)       12  Screen 9c:  Typical decision screen for four-box Discoordination game (frame 2b)   Screen 9d:  Typical decision screen for eight-box Discoordination game (frame 2b)       13  Screen 9e:  Typical decision screen for hider in four-box Hide and Seek game (frame 1a)  Screen 9f:  Typical decision screen for hider in eight-box Hide and Seek game (frame 2b)      14   Screen 9g:  Typical decision screen for seeker in four-box Hide and Seek game (frame 1a)  Screen 9h:  Typical decision screen for seeker in eight-box Hide and Seek game (frame 2b)     15    Screen 10a:  Introduction to part 2: CD sessions  Screen 10b:  Introduction to part 2: HS sessions     16   Screen 11a: Reminder of instruction in the new role: hider  Screen 11b: Reminder of instruction in the new role: seeker  Note: instruction screens for Coordination and Discoordination games in the second part of CD sessions are the same as screens 6a and 6b respectively.    17   Screen 12:  Typical outcome screen: CD and HS sessions  Note: the example shown above is for an HS session.  In CD sessions, the screen makes no reference to roles.  Screen 13:  typical payment screen: CD and HS sessions    18  III.  Additional analysis of position effects In Section 3.1 of the main paper, we claimed that position was not a major determinant of players’ choices.  In support of this claim, we reported the frequencies with which subjects’ chosen boxes were in each of the four or eight possible positions (Table 2 in the main paper).  As a further robustness check, we consider whether the position of the oddity influenced the likelihood of its being chosen.  Table A2 reports the relative frequency with which the oddity was chosen, conditional on its position.  Under the null hypothesis that choices are not influenced by position , the expected relative frequency of oddity choices is the same for all oddity positions.  For each combination of player role and type of game, Table 4 in the main paper reports 95 per cent confidence intervals for the probability of oddity choices, derived from the aggregate data.  Cases in which, for a specific position, the observed relative frequency of oddity choices lies outside these confidence intervals are shown in bold in Table A2.  As with the data reported in Table 2, deviations from randomness are relatively small and show no obvious pattern.   TABLE A2: RELATIVE FREQUENCY OF ODDITY CHOICES BY POSITION OF ODDITY Percentage of oddity choices when oddity is in position:  role    1  2  3  4  5  6  7  8 Avg. _____________________________________________________________________________________four- box games coordinators (n = 40)  76.4  73.5  77.7  76.0     76.0 discoordinators (n = 40) 23.2  19.1  23.0  28.3     23.3 hiders (part 1: n = 100)  10.1  15.3  17.7  15.5     14.6 hiders (part 2: n = 100)  17.9  18.1  20.7  15.8     18.2 seekers (part 1: n = 100)  24.1  30.7  27.9  21.0     25.8 seekers (part 2: n = 100)  22.3  20.5  22.0  18.1     20.8  eight- box games coordinators (n = 40)  73.0  81.7  72.4  80.6  81.2  66.7  78.3  73.1 76.0 discoordinators (n = 40)  11.5  15.3  24.4  18.1    9.4  11.0  19.3  12.1 15.4 hiders (part 1: n = 100)    8.3    8.8    9.3    9.3    5.7    6.9    7.6  12.8   8.6 hiders (part 2: n = 100)  14.2  11.3    8.2    7.9  11.2  11.1    8.5    6.5   9.8 seekers (part 1: n = 100)  25.7  31.0  34.6  22.9  22.0  24.5  19.3  23.9    25.3 seekers (part 2: n = 100)  17.0  16.1  23.5  17.5  20.8  16.2    8.5  11.4   16.1 _____________________________________________________________________________________ 19  IV.  Analysis restricted to games with neutral connotations  One difference between our analysis and CI’s is that CI restrict their analysis to Hide and Seek games in which the oddity had neutral connotations, while our experiment also included games in which those connotations were positive or negative.  In this Section, we consider whether our results would be affected if we restricted our analysis to neutrally framed games (i.e. frames 4a, 4b, 5a, 5b, 6a, 6b, 13a, 13b, 14a, 14b, 15a and 15b).  Tables A3 and A4 reproduce the analyses in Tables 2 and 4 respectively of the main paper, but using only data from neutral frames.  Table A3 shows that, for each class of games and for each role, each position was chosen with approximately the same frequency; there are a few instances of statistically significant differences from the random-choice benchmark, but these do not seem to follow any obvious pattern.  Thus, the conclusion we drew from Table 2 – that position effects are relatively unimportant – is not an artifact of the pooling of data across games in which oddities have different connotations. Table A4 reports the frequency of oddity choices by players in each role.  The observed frequencies are generally very similar to those in Table 2, but because they are derived from fewer game, the confidence intervals are wider.  The restricted data, just like the unrestricted, clearly disconfirm Implication 1 (for each of the three roles of discoordinator, hider and seeker, and for both four- and eight-box games, the upper bound of the 95 per cent confidence interval of the frequency of oddity choices is less than 0.33).  As with the unrestricted data, the observed proportion of oddity choices by hiders is significantly higher in four-box games than in eight-box games, contrary to Implication 2, but there are no significant differences for coordinators (as in the unrestricted data), for seekers (as in the unrestricted data), or for discoordinators.  In the light of these results, we see no reason to think that the main conclusions of our paper are attributable to our use of frames with positive and negative connotations.    20  TABLE A3:  FREQUENCY OF CHOICES BY POSITION (NEUTRAL FRAMES ONLY)              percentage of choices that are of position: role    1 2 3 4 5 6 7 8 ___________________________________________________________________________ four- box games coordinators (n = 40)  31.2* 23.7 25.8 19.2 discoordinators (n = 40) 26.7** 21.7 28.0 23.7 hiders (part 1: n = 100)  24.3 20.0 25.3 30.3** hiders (part 2: n = 100)   25.0 23.7* 26.0 25.3 seekers (part 1: n = 100) 21.3 30.3 27.0 21.3 seekers (part 2: n = 100) 23.3 24.0 33.7*** 19.0* eight- box games coordinators (n = 40)  12.1 15.0   5.8*** 11.2 18.3**   9.6 17.9* 10.0*  discoordinators (n = 40)     9.6 12.9 13.7 13.3 14.6 12.1 12.5 11.2 hiders (part 1: n = 100)  10.7 16.7* 11.7 10.3 12.0 14.0 14.7 10.0  hiders (part 2: n = 100)   12.7 14.0 13.0 12.0 12.3 17.3** 10.3   8.3* seekers (part 1: n = 100)  9.3 12.7 14.7 14.3 15.0 12.0 10.3 11.7 seekers (part 2: n = 100) 11.3 10.7 15.0 11.3 18.0** 11.3 12.0*** 10.3 _____________________________________________________________________ Note: n denotes the number of subjects who faced games of the relevant type.  Each coordinator and discoordinator faced 6 games of that type; each hider and seeker faced 3 games of that type.   Relative frequencies greater than the random-choice benchmark (25 per cent for four-box games, 12.5 per cent for eight-box games) are shown in bold.  For each type of game and each position j, we find the number of choices made in position j by each subject i, and then run a chi-squared test of whether the distribution of these n numbers is different from the binomial distribution implied by random choice.  Significant differences are shown by asterisks (*, ** and *** denoting significance at the 10, 5 and 1 per cent levels).    21  TABLE A4:  FREQUENCY OF ODDITY CHOICES BY PLAYERS IN ALL ROLES (NEUTRAL FRAMES ONLY)      percentage of all choices that are of oddity:   role    lower bound observed upper bound   _______________________________________________________________  four-box games coordinators (n = 40)   72.0  80.8  89.7   discoordinators (n = 40) 13.2  20.8  28.4     hiders (part 1: n = 100)  10.3  15.0  19.7     hiders (part 2: n = 100)  13.7  19.0  24.3     seekers (part 1: n = 100) 17.1  22.7  28.2     seekers (part 2: n = 100) 14.4  19.7  24.9      eight-box games coordinators (n = 40)  76.6  84.2  91.7   discoordinators (n = 40)  11.1   18.7  26.4  hiders (part 1: n = 100)    3.9    8.0  12.0  hiders (part 2: n = 100)    3.6    7.3  11.0 seekers (part 1: n = 100) 17.3   22.3   27.4     seekers (part 2: n = 100) 10.2   15.3   20.4   _______________________________________________________________ Note: n denotes the number of subjects who faced games of the relevant type.  Each coordinator and discoordinator faced 6 games of that type.  Each hider and seeker faced 3 games of that type.   22  V.  Patterns in Hide and Seek data consistent with ‘plausible’ level-k models This Section uses the same notation and assumptions as are used in Sections 1 and 4 of the main paper.   Consider a large population of potential players of the ABAA game.  Let ρBH and ρCH be the probabilities with which hiders choose B and central A respectively.  We define ρEH  = (1 – ρBH – ρCH)/2, i.e. the average probability with which hiders choose end labels.  Ignoring ties for simplicity, we define the modal choice of hiders mH so that it takes the value B, C or E according to which of ρBH,  ρCH and  ρEH is greatest.  The modal choice of seekers mS is defined analogously.  We define a pattern as a triple (mH, mS, z) where z takes the value H if mH > mS and S if mS > mH.  There are eighteen possible patterns.  One of these, namely (C, C, S), is the fatal attraction pattern.    Our representation of CI’s concept of a plausible level-k model is explained in Section 4 of the main paper.  We treat the following assumptions about the population distribution of types as necessary properties of a plausible model:    (1)   π1, π2, π3, π4 ≥ 0  (2)  π1 + π2 +π3 + π4 = 1 (3) π2 > π1 (4)   π3 > π4.  Two alternative assumptions about salience are treated as plausible: either B is most salient (Specification 1), or the end As are most salient (Specification 2).  We consider these in turn.  Specification 1: B is most salient  Level-k theory implies the following choice probabilities: (5)   ρBH  = π3/3 + π4 (6) ρCH  = π1 + π2/3 (7) ρEH  = π2/3 + π3/3 (8) ρBS  = π1 + π4/3 (9) ρCS  = π2 + π3/3 (10) ρES  = π3/3 + π4/3  Expressions (1) to (10) imply the following results: 23  Result 1(i):  Not [mS = B].  Proof:  Suppose mS = B.  Then ρBS > ρCS  which, by (8) and (9), implies π1 + π4/3 > π2 + π3/3.  But by (3) and (4), π1 + π4/3 <  π2 + π3/3, a contradiction. Result 1(ii):  Not [mH = C and mS = E].  Proof:  Suppose mH = C and mS = E.  Then ρCH > ρEH  which, by (6) and (7) implies π3/3 < π1.  But also ρES  > ρBS which, by (8) and (10), implies π1 < π3/3, a contradiction. Result 1(iii):  Not [mH = E and mS = E].  Proof:  Suppose mH = E and mS = E.  Then ρEH  > ρBH which, by (5) and (7), implies π2/3 > π4.  But also ρES  > ρCS which, by (9) and (10), implies π4/3 > π2.  Adding these inequalities gives (π2 + π4)/3 > π2 + π4.  Given (1), this is a contradiction.  Result 1(iv):  Not [mS = E and z = S].  Proof: suppose [mS = E and z = S].  Then ρES > ρBH which, by (5) and (10), implies π4/3 > π4.  Given (1), this is a contradiction. Result 1(v):  Not [mH = E and z = H].  Proof: suppose [mH = E and z = H].  Then ρEH  > ρCS which, by (7) and (9), this implies π2/3 > π2.  Given (1), this is a contradiction.  Of the eighteen possible patterns, only six are consistent with Results 1(i) to 1(v).  These are listed below.  For each pattern, we give an example of a ‘plausible’ distribution of types that induces that pattern.  Pattern 1: (B, C, H).  Example: π1 = 0.05, π2 = 0.15, π3 = 0.51, π4 = 0.29. Pattern 2: (B, C, S).  Example: π1 = 0.05, π2 = 0.36, π3 = 0.30, π4 = 0.29. Pattern 3: (B, E, H).  Example: π1 = 0.04, π2 = 0.06, π3 = 0.50, π4 = 0.40. Pattern 4: (C, C, H).  Example: π1 = 0.35, π2 = 0.40, π3 = 0.15, π4 =  0.10.   Pattern 5: (C, C, S).  Example: π1 = 0.20, π2 = 0.40, π3 = 0.30, π4 =  0.10. Pattern 6: (E, C, S).  Example: π1 = 0.10, π2 = 0.13, π3 = 0.73, π4 = 0.04.   Specification 2: End As are most salient Level-k theory implies the following choice probabilities: (11)   ρBH  = π2/2 + π3/3 (12) ρCH  = π1 + π2/2 (13) ρEH  = π3/3 + π4/2 (14) ρBS   = π3/2 + π4/3 (15) ρCS   = π2 + π3/2 24  (16) ρES   = π1/2 + π4/3 Expressions (1) to (4) and (11) to (16) imply the following results: Result 2(i):  Not [mS = E].   Proof: suppose mS  = E.  Then ρES  > ρCS which, by (15) and (16), implies π1/2 + π4/3 > π2 + π3/2.  But by (1), (3) and (4), 0 ≤ π1 < π2 and 0 ≤ π4 < π3.  Thus π1/2 < π2 and π4/3 < π3/2.  Summing these inequalities gives π1/2 + π4/3 < π2 + π3/2, a contradiction. Result 2(ii):  Not [mH = B and mS = B].  Proof:  Suppose mH = B and mS = B.  Then ρBH > ρEH  which, by (11) and (13), implies π2 > π4.  But also ρBS > ρCS which, by (14) and (15), implies π4/3 > π2.  Given (1), this implies π4 > π2, a contradiction. Result 2(iii):  Not [mH = C and mS = B].  Proof:  Suppose mH = C and mS = B.  Then ρCH > ρBH  which, by (11) and (12), implies π1 > π3/3.  But also ρBS > ρCS which, by (14) and (15), implies π4/3 > π2.  Adding these inequalities gives π1 + π4/3 > π3/3 + π2.  But by (3) and (4), π1 < π2 and π4 < π3, a contradiction. Result 2(iv):  Not [mH = E and z = H].  Proof: suppose [mH = E and z = H].  Then ρEH  > ρBS which, by (13) and (14), implies π4 > π3.  Given (4), this is a contradiction. Result 2(v):  Not [mH = B and mS = C and z = H].  Proof: suppose mH = B and mS = C and z = B.  Then BH > CS which, by (11) and (15), implies π2/2 + π3/6 < 0, contrary to (1): a contradiction.  Of the eighteen possible patterns, only five are consistent with Results 2(i) to 2(v).  These are listed below.  For the one pattern that is not also consistent with Case 1, we give an example of a ‘plausible’ distribution of types that induces that pattern: Pattern 2: (B, C, S).  Also consistent with Case 1. Pattern 4: (C, C, H).  Also consistent with Case 1. Pattern 5: (C, C, S).  Also consistent with Case 1. Pattern 6: (E, C, S).  Also consistent with Case 1. Pattern 7:  (E, B, S).  Example: π1 = 0.04, π2 = 0.06, π3 = 0.50, π4 = 0.40.  We have now established that exactly seven of the eighteen possible patterns are consistent with ‘plausible’ level-k models.    

How Portable is Level-0 Behavior? A Test of Level-k Theory in Games with Non-Neutral Frames

We test the portability of level-0 assumptions in level-k theory in an experimental investigation of behavior in Coordination, Discoordination, and Hide and Seek games with common, non-neutral frames. Assuming that level-0 behavior depends only on the frame, we derive hypotheses that are independent of prior assumptions about salience. Those hypotheses are not confirmed. Our findings contrast with previous research which has fitted parameterized level-k models to Hide and Seek data. We show that, as a criterion of successful explanation, the existence of a plausible model that replicates the main patterns in these data has a high probability of false positives

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How portable is level-0 behavior?:A test of level-k theory in games with non-neutral frames

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How portable is level-0 behavior? A test of level-k theory in game with non-neutral frames

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