1 School of Economics University of East Anglia Norwich NR4 7TJ, United Kingdom Focal points in tacit bargaining games Andrea Isoni #, Anders Poulsen *, Robert Sugden * and Kei Tsutsui* 27 February 2010 # Department of Economics, University of Warwick * Centre for Experimental and Behavioural Social Science, University of East Anglia Work in progress with provisional data analysis: please do not quote without permission of authors.

2 Schelling’s theory of focal points (Strategy of Conflict, 1960) Starts with pure coordination games (e.g. ‘Choose a place and time of day to meet the other person in New York city’, ‘Choose Heads or Tails’). These games have symmetrical payoffs, e.g. Player B HeadsTails Player AHeads1, 10, 0 Tails0, 01, 1 Schelling presents ‘unscientific evidence’ that ordinary players coordinate successfully by finding ‘focal points’ (e.g. 86% choose Heads). This finding confirmed in controlled experiments (Mehta et al, AER, 1994).

3 But for Schelling, pure coordination games are only a model. He argues that focal points are used to resolve bargaining problems (e.g. Cold War conflicts). He generalises from pure coordination games to tacit bargaining problems (= coordination games with payoff asymmetries), e.g. the parachutists’ game, which I simplify to: Player B (nearer tree) BridgeTree Player ABridge10, 90, 0 (nearer bridge)Tree0, 09, 10 He claims that focal points can resolve these games (and that classic bargaining solutions, e.g. Nash’s, are merely focal points ‘limited to the universe of mathematics’).

4 And then he generalises to explicit bargaining games (i.e. with communication), and argues that focal points can resolve these games too. (A ‘cheap talk’ argument: if there really is common knowledge that one equilibrium is focal and the others aren’t, that knowledge can’t be removed by talking. ‘Beggars cannot be choosers when fortune gives the signals’.) If Schelling is right, bargaining theory needs to take account of focality. Up to now (for the most part) it hasn’t. But is he right? Most focal point experiments (there haven’t been many) have investigated pure coordination games. But a recent exception...

5 Vincent Crawford, Uri Gneezy and Yuval Rottenstreich, ‘The power of focal points is limited: even minute payoff asymmetry may yield large coordination failures’ (AER 2008) Crawford et al use the salience of ‘X’ relative to ‘Y’: Player 2 (‘favoured’) XY Player 1 X 5, 5+a0, 0 (‘unfavoured’)Y0, 05+a, 5 (payoffs in US dollars) ‘symmetry’: a = 0 ‘slight asymmetry’: a = 0.1 ‘moderate asymmetry’: a = 1 ‘large asymmetry’: a = 5 Does asymmetry affect ‘power of focal points’?

6 Crawford et al’s results: a=0a=0.1a=1a=5 (n=50)(n=48)(n=61)(n=21) _____________________________________________________ % choosing X: unfavoured (X = choose 5)n/a favoured (X = choose 5+a)n/a all coordination index: observed (mixed-strategy equilibrium)(50)(50)(50)(44) ______________________________________________________ Notice: 1. ‘X’ is focal when payoffs symmetrical, but not otherwise 2. both players tend to choose 5 when asymmetry very small (a = 0.1) – which induces discoordination 3. both players tend to choose 5+a when asymmetry is significant (a = 1 or a = 5) – which also induces discoordination.

7 (My) interpretation: in the symmetrical game, players can immediately see that they need to coordinate on the same strategy. The payoff information doesn’t help them to do this, so they look for some ‘clue’ that can help. The only clue is the difference between ‘X’ and ‘Y’; so they use that. (This is the kind of ‘joint’ reasoning Schelling describes.) But in the asymmetrical games, it’s less obvious that the ‘real’ problem is to coordinate. The payoff information looks relevant, while (at first sight) the difference between ‘X’ and ‘Y’ doesn’t. Each player starts thinking ‘Will the other player go for the higher or lower payoff?’, ‘What will he expect me to do?’, etc. (This is ‘individual’ reasoning.) This is a one-shot game, so we shouldn’t expect the mix of strategies to be a Nash equilibrium. So, these results raise doubts about the relevance of focal points for asymmetrical games (and therefore for bargaining problems).

8 But... Consider the framing of Crawford et al’s games. A typical asymmetrical game was presented as [check exact wording]: Choose X or Y. If you both choose X, P1 receives $5 and P2 receives $6. If you both choose Y, P1 receives $6 and P2 receives $5. If you choose differently, neither of you receives anything. This framing doesn’t suggest bargaining... The problem is presented as a choice between distributions of money between the players; in bargaining, one makes claims or demands, to which the other person may or may not accede. The labelling ‘clue’ distinguishes the ($5, $6) distribution, but it doesn’t connect specific players to specific objects of value – it doesn’t suggest that the $5 ‘belongs to’ P1 and the $6 ‘belongs to’ P2.

9 6 5 Contrast the ‘bargaining table’ display: One player is Blue, the other is Red. Each has a base (the coloured square). The discs have the money values shown (in £). The players simultaneously make claims on discs by clicking on them to connect them to their bases...

If no disc is claimed by both, each gets the discs he/she has claimed; but if any disc is claimed by both, both get nothing.

11 The normal form of the game is: Red player nonenearfarboth none0, 00, 50, 60, 11 Bluenear6, 06, 50, 00, 0 playerfar5, 00, 05, 60, 0 both11, 00, 00, 00, 0 ‘none’ = claim no discs ‘near’ = claim the disc closer to own base (=6 for Blue, 5 for Red) ‘far’ = claim the disc further from own base (=5 for Blue, 6 for Red) ‘both’ = claim both discs Notice that payoff matrix is symmetrical except for labelling. But ‘none’ is weakly dominated by every other strategy. Deleting that...

12 after deleting ‘none’ Red player nearfarboth Bluenear6, 50, 00, 0 playerfar0, 05, 60, 0 both0, 00, 00, 0 Now ‘both’ is weakly dominated by every other strategy. Deleting that...

13 After deleting ‘both’ Red player nearfar Bluenear6, 50, 0 playerfar0, 05, 6 This is equivalent to Crawford et al’s asymmetric game, but with a different framing, more suggestive of bargaining. (Note similarity to Nash demand game.) (The ‘iterated deletion’ argument seems intuitive: the only way to earn anything is by claiming discs, so there’s no point in not claiming anything; since the other player will surely realise this, claiming both discs is pointless too.)

14 Our experiment Each subject plays 24 ‘bargaining table’ games against an anonymous opponent. In each game, the subject records his claims; no information about the other player’s claims is given until the end of the experiment. Games presented in random order. At the end of the experiment, one game chosen at random for the pair of players; each paid according to the result of this game.

15 Games are of four types, defined by a 2x2 classification: Symmetrical/ asymetrical In symmetrical games, the total value of the discs is £10, and a £5:£5 division is possible. In asymmetrical games, the total value of the discs is £11, and an equal division is not possible. Usually the most equal possible division is £6:£5, but sometimes it is £8:£3 or £10:£1. Spatial cues/ no spatial cues In game with spatial cues, one division of the discs (always one of the most equal possible) is ‘suggested’ by the spatial layout of discs relative to bases. In asymmetrical games, this implies that one player is favoured, the other unfavoured. In games without spatial cues, the spatial layout doesn’t suggest any particular division (or: is intended not to suggest...).

(This is Game 2. Summary description: 5||5) Symmetrical (a £5:£5 division is possible) Spatial cues (suggested division of discs by closeness: Blue takes top left disc, Red takes bottom right disc) Four examples:

(This is Game 1. Summary description: 6||5) Asymmetrical (the most equal possible division is £6:£5) Spatial cues (suggested division of discs by closeness: Blue takes top left £5 disc, Red takes bottom right £6 disc)

(This is Game 4. Summary description: |5,5|) Symmetrical (a £5:£5 division is possible) No spatial cues

(This is Game 3. Summary description: |6,5|) Asymmetrical (the most equal possible division is £6:£5) No spatial cues

20 Note on cues: in games which we have classified as having ‘no spatial cues’, it is still possible for players to distinguish particular divisions. Consider rules: 1. (In asymmetrical game) Red claims £6, Blue claims £5 2. (In asymmetrical game) Player on left claims £6, player on right claims £5 3. Player on left takes top disc, player on right takes bottom disc. Any of these might be perceived as focal. This is an inescapable feature of coordination games. If a game genuinely has two or more equilibria, then players must be able to distinguish them, i.e. the equilibria must have distinct labels. ‘Focality’ is a property of the content of the labels. But, we control for this in two ways...

21 First control: Each game is displayed (to different subsets of subjects) in four different ways, using left/right and red/blue transpositions. This allows us to test whether subjects use left/right or red/blue distinctions as a means of coordinating. (Preliminary analysis suggests they don’t.) Second control: We compare matched games with and without ‘spatial cues’. This allows us to test whether, other things being equal, the presence of ‘spatial cues’ (as we define them) increases the extent of coordination. If they do, this is evidence that focal points are effective in tacit coordination games.

22 Some results: 1. Do subjects use spatial cues? Various tests are possible, but all give answer: overwhelmingly, YES. E.g. consider each game with spatial cues, and each pair of ‘identical twin’ discs such that one disc is closer to one player’s base, the other to the other’s. Is the closer disc claimed more frequently than the less close one? In all 18 cases, yes (significant at 1% level). [In all 6 cases of ‘non-identical twins’, i.e. twins by location but one disc has higher value than the other, the other-things-equal answer is also ‘yes’, significant at at least 5% level.] E.g. aggregating over all ‘identical twin’ cases, how does the frequency with which discs are chosen vary with disc location (defined by column)?...

column numbers (standardised for player on left) frequency with which disc chosen by closer player (%) (notice discontinuity at centre line)

24 2. Average outcomes Here (as a first step) we report average actual outcomes for players (i.e. pooling across left/right and red/blue transpositions and pairing each player with his/her actual opponent). Efficiency = (sum of outcomes for both players)/(total value of discs on table) for asymmetrical games with spatial cues: Outcome asymmetry = (outcome for favoured player)/(outcome for unfavoured player) If (in otherwise matched games) efficiency is greater when there are spatial cues, this is evidence that focal points are effective. We are particularly interested in whether this effect occurs for asymmetric games. If there are outcome asymmetries (with favoured players getting higher payoffs), this is further evidence that focal points have real consequences in asymmetrical games.

25 (i) Games 1, 2, 3, 4 (2 discs, 5:5 or 6:5) These are the four games you’ve already seen, but as a reminder:

Game 2 (symmetrical, spatial cues)

Game 1: asymmetrical, spatial cues

Game 4: symmetrical, no spatial cues

Game 3: asymmetrical, no spatial cues

30 Results for games 1, 2, 3, 4 (2 discs, 5:5 or 6:5) Efficiency (%): symmetrical asymmetrical spatial cues8361 no spatial cues5250 Outcome asymmetry in asymmetrical game with cues: 1.14 Interpretation: spatial cues increase efficiency in both symmetrical and asymmetrical games; but effect much stronger for symmetrical games (a weaker version of Crawford et al’s results); in the asymmetrical game with spatial cues, the outcome is slightly skewed towards the favoured player (compare 6/5 = 1.20).

31 In the preceding asymmetrical games, the two discs were 6 and 5. But we also had 2-disc games with greater payoff asymmetry...

32 (ii) Games 1, 2, 13, 15 (2 discs, 5:5 or 8:3) Efficiency (%): symmetrical asymmetrical spatial cues8346 no spatial cues5220 Outcome asymmetry in asymmetrical game with cues: 1.69 Interpretation: spatial cues increase efficiency in both symmetrical and asymmetrical games (baseline efficiency in asymmetrical game is relatively low because of tendency for players to claim 8 rather than 3 [compare Nash equilibrium efficiency = 40%]); in the asymmetrical game with spatial cues, the outcome is skewed towards the favoured player (compare 8/3 = 2.67).

33 (iii) Games 1, 2, 14, 16 (2 discs, 5:5 or 10:1) Efficiency (%): symmetrical asymmetrical spatial cues8344 no spatial cues5219 Outcome asymmetry in asymmetrical game with cues: 2.23 Interpretation: spatial cues increase efficiency in both symmetrical and asymmetrical games (baseline efficiency in asymmetrical game is relatively low because of tendency for players to claim 10 rather than 1 [compare Nash equilibrium efficiency = 17%]); in the asymmetrical game with spatial cues, the outcome is skewed towards the favoured player (compare 10/1 = 10.0).

34 In the preceding games, there were only two discs. But we also had games with 4 discs and with 8 discs. Here are the basic four-disc games:

Game 6 (symmetrical, spatial cues) 2 2

Game 5 (asymmetrical, spatial cues) 2 3

Game 12 (symmetrical, no spatial cues) 2 2

Game 11 (asymmetrical, no spatial cues) 2 3

39 (iv) Games 5, 6, 11, 12 (4 discs, 5:5 or 6:5) Efficiency (%): symmetrical asymmetrical spatial cues6357 no spatial cues3724 Outcome asymmetry in asymmetrical game with cues: 1.00 Interpretation: spatial cues increase efficiency in both symmetrical and asymmetrical games (baseline efficiency lower with 4 discs than with 2); in the asymmetrical game with spatial cues, the outcome is not skewed; symmetrical and asymmetrical games seem to be converging.

40 Now here are the basic 8-disc games:

41 1 Game 18 (symmetrical, spatial cues)

42 2 Game 17 (asymmetrical, spatial cues)

43 2 Game 20 (symmetrical, no spatial cues)

44 2 Game 19 (asymmetrical, no spatial cues)

45 (v) Games 17, 18, 19, 20 (8 discs, 5:5 or 6:5) Efficiency (%): symmetrical asymmetrical spatial cues5351 no spatial cues1514 Outcome asymmetry in symmetrical game with cues: 1.08 Interpretation: spatial cues increase efficiency in both symmetrical and asymmetrical games (baseline efficiency even lower with 8 discs); in the asymmetrical game with spatial cues, the outcome is slightly skewed (but perhaps not significantly?); symmetrical and asymmetrical games seem to have converged.

46 Conclusions I haven’t had time to report all the games (the games I haven’t reported investigate the effect of arranging discs in ‘blocks’). But the general patterns I’ve shown you appear throughout the experiment...

47 1. In all games, players use spatial cues (i.e. tend to claim discs that are relatively close to their own base). 2. Because of this, there is a very strong and very general tendency for efficiency to be greater in games with spatial cues than in matched games without those cues. 3. When there are only two discs and the payoff asymmetry in asymmetrical games is relatively low (i.e. 6:5), we find a weaker version of Crawford et al’s result, i.e. focal points have more power in symmetrical games than in asymmetrical ones. 4. But this ‘relative power’ effect dissipates if the payoff asymmetry is greater (8:3 or 10:1) – though unsurprisingly, efficiency is lower in asymmetrical games. 5. The relative power effect dissipates if the number of discs is increased.

48 Summing up: Our results broadly support Schelling’s hypothesis that behaviour in tacit bargaining games is influenced by focal points. Our next task (we are currently designing a suitable experiment) is to investigate whether the effects we have found extend to explicit bargaining games.

49 Thank you for listening.

50 Appendix 1: all games in basic form

Game 1 6||5 asymmetric payoffs (most equal efficient division 6:5) spatial cues 2 discs

Game 2 5||5 symmetric payoffs (most equal efficient division 5:5) spatial cues 2 discs

Game 3 |6,5| asymmetric payoffs (most equal efficient division 6:5) no spatial cues 2 discs

Game 4 |5,5| symmetric payoffs (most equal efficient division 5:5) no spatial cues 2 discs

Game 5 3,3||3,2 asymmetric payoffs (most equal efficient division 6:5) spatial cues 4 discs 2 3

Game 6 3,2||3,2 symmetric payoffs (most equal efficient division 6:5) spatial cues 4 discs 2 2

Game 7 (3,3)//(3,2) asymmetric payoffs (most equal efficient division 6:5) spatial cues with blocks 4 discs 2 3

Game 8 (3,2)//(3,2) symmetric payoffs (most equal efficient division 5:5) spatial cues with blocks 4 discs 2 2

Game 9 |(3,3)(3,2)| asymmetric payoffs (most equal efficient division 6:5) no spatial cues; blocks 4 discs 2 3

Game 10 |(3,2)(3,2)| symmetric payoffs (most equal efficient division 5:5) no spatial cues; blocks 4 discs 2 2

Game 11 |3,3,3,2| asymmetric payoffs (most equal efficient division 6:5) no spatial cues; no blocks 4 discs 2 3

Game 12 |3,3,2,2| symmetric payoffs (most equal efficient division 5:5) no spatial cues; no blocks 4 discs 2 2

Game 11 |3,3,3,2| asymmetric payoffs (most equal efficient division 6:5) no spatial cues; no blocks 4 discs 3 3

Game 12 |3,3,2,2| symmetric payoffs (most equal efficient division 5:5) no spatial cues; no blocks 4 discs 2 3

Game 13 8||3 asymmetric payoffs (most equal efficient division 8:3) spatial cues 2 discs

Game 14 10||1 asymmetric payoffs (most equal efficient division 10:1) spatial cues 2 discs

Game 15 |8,3| asymmetric payoffs (most equal efficient division 8:3) no spatial cues 2 discs

Game 16 |10, 1| asymmetric payoffs (most equal efficient division 10:1) no spatial cues 2 discs

69 2 Game 17 2,2,1,1||2,1,1,1 asymmetric payoffs (most equal efficient division 6:5) spatial cues 8 discs

70 1 Game 18 2,1,1,1||2,1,1,1 symmetric payoffs (most equal efficient division 5:5) spatial cues 8 discs

71 2 Game 19 |2,2,2,1,1,1,1,1| asymmetric payoffs (most equal efficient division 6:5) no spatial cues; no blocks 8 discs

72 2 Game 20 |2,2,1,1,1,1,1,1| symmetric payoffs (most equal efficient division 5:5) no spatial cues; no blocks 8 discs

73 2 Game 21 (2,2,1,1)//(2,1,1,1) asymmetric payoffs (most equal efficient division 6:5) spatial cues with blocks 8 discs

74 2 Game 22 (2,1,1,1)//(2,1,1,1) symmetric payoffs (most equal efficient division 5:5) spatial cues with blocks 8 discs

75 2 Game 23 |(2,2,1,1)(2,1,1,1)| asymmetric payoffs (most equal efficient division 6:5) no spatial cues; blocks 8 discs

76 2 Game 24 |(2,1,1,1)(2,1,1,1)| symmetric payoffs (most equal efficient division 5:5) no spatial cues; blocks 8 discs

77 Appendix 2: transpositions

Game 1 [6||5] in basic form Blue at left Higher-valued disc to top and left

Game 1 with Red/Blue transposition Red at left Higher-valued disc to top and left

Game 1 with diagonal transposition of discs (note: this has no effect on symmetrical games, except Game 22) Blue at left Higher-valued disc to bottom and right

Game 1 with Red/Blue transposition and diagonal transposition of discs Red at left Higher-valued disc to bottom and right

82 The end