Two Person Bargaining Mechanisms: A Laboratory Experiment

Abstract

We conduct a series of experiments in which two subjects bargain over five options. Following an experimental design closely related to De Clippel et al. (Am Econ Rev 104:3434–3458, 2014), we evaluate the performance of three bargaining mechanisms: (α) one subject shortlists a block of three options before the other chooses one among them, ($\beta$) both subjects veto options simultaneously and in a block, and ($\gamma$) both subjects veto options simultaneously and gradually one after the other. We document that the non-symmetric shortlisting mechanism (α) is highly efficient, but our data also suggest the existence of a first-mover advantage as subjects become more experienced. The simultaneous mechanism ($\beta$) is less efficient than (α) and generates a high level of ex-post inequality. The gradual veto mechanism ($\gamma$) is no less efficient than (α), but has the important advantage of shutting down the possibility of any first-mover advantage.

Appendices

Appendix

A. Equilibria of the Gradual Vetoes Mechanism

This section shows that the subgame-perfect equilibrium outcomes of the gradual vetoes mechanism coincide with the FB options in the profiles used in the experiment. The game we are considering is an extensive form game with complete information and simultaneous moves, and the definition of subgame-perfect equilibrium is standard: it is a strategy profile that generates a Nash equilibrium in each subgame.

The main result of this section holds under two assumptions: (i) players only use weakly undominated strategies and (ii) their preferences over lotteries satisfy stochastic dominance. Ruling out weakly undominated strategies implies that, at each stage, no player casts a veto against her best available option on the stage. Stochastic dominance is satisfied by most preference extensions in the literature. It simply requires that, when comparing two lotteries, a shift in probability to strictly preferred options yields strictly preferred lotteries. As in the case of subgame perfect equilibrium, we do not include its formal definition since it is standard.

Before laying out the analysis of the four profiles, we fully describe the outcomes with only two and three options since this will be useful in the sequel.

• 1. Two options

With two options, only four preference profiles are possible: (i) $a \succ_{i} b$ for $i$ = 1, 2, (ii) $b \succ_{i} a$ for $i$ = 1, 2, (iii) $a \succ_{1} b$ and $b \succ_{2} a$ and (iv) $b \succ_{1} a$ and $a \succ_{2} b$. Since every player vetoes her worst option since this the only weakly non-dominated strategy, it follows that in profiles (i) and (ii), the only equilibrium outcome is the unanimously preferred option whereas in profiles (iii) and (iv) the only outcome is the lottery $uni\left( {a, b} \right).$

• 2. Three options

Each equilibrium selects a unique FB option and the set of equilibrium outcomes corresponds to the set of FB options. W.l.o.g we fix $a$ ≻₁ $b$ ≻₁ $c$ and consider the different preferences of player 2. This is possible since the game is symmetric so permuting the name of the options would give the corresponding payoff matrix and equilibrium outcome.

When $a$ ≻₂ $b$ ≻₂ $c$ and when $a$ ≻₂ $c$ ≻₂ $b$, $a$ is the unique equilibrium outcome since the payoff matrix equals:

	b	c
b	a	a
c	a	a

When $b$ ≻₂ $a$ ≻₂ $c$, both $a$ and $b$ are equilibrium outcomes whereas $b$ is the only equilibrium outcome when b ≻₂ c ≻₂ a as one deduces from the payoff matrix:

	a	c
b	c	a
c	b	uni(a,b)

Finally, when $c$ ≻₂ $a$ ≻₂ $b$, the unique equilibrium outcome is $a$ whereas when $c$ ≻₂ $b$ ≻₂ $a$, only b is selected in equilibrium as one can see in the next payoff matrix:

	a	b
b	c	uni(a,c)
c	b	a

• 3. Profile A0

In this profile $a$ ≻₁ $b$ ≻₁ $c$ ≻₁ $d$ ≻₁ $e$ and $e$ ≻₂ $d$ ≻₂ $c$ ≻₂ $b$ ≻₂ $a$. The matrix of the game is as follows:

	a	b	c	d
b	d	${\Gamma }\left( {a,c,d,e} \right)$	d	c
c	d	d	${\Gamma }\left( {a,b,d,e} \right)$	b
d	c	c	b	${\Gamma }\left( {a,b,c,e} \right)$
e	c*	c	b	b

It follows that $c$ is the unique equilibrium outcome with the strategy $\left( {e, a} \right)$ being an equilibrium. This outcome is independent of the outcomes of the subgames that involve 4 options, analyzed in the sequel.

$\Gamma \left( {a,c,d,e} \right)$: This game admits a unique equilibrium (e, a) with outcome $uni\left( {c, d} \right)$ with preferences $a$ ≻₁ $c$ ≻₁ $d$ ≻₁ $e$ and $e$ ≻₂ $d$ ≻₂ $c$ ≻₂ $a$. Its matrix is as follows.

	a	d	e
c	d or e	d	uni(a,e)
d	uni(c,e)	uni(a,e)	c
e	uni(c,d)	uni(a,d)	uni(a,c)

$\Gamma \left( {a,b,d,e} \right)$: This game admits a unique equilibrium $\left( {e, a} \right)$ with outcome $uni\left( {b, d} \right)$ since restricted preferences are given by $a$ ≻₁ $b$ ≻₁ $d$ ≻₁ $e$ and $e$ ≻₂ $d$ ≻₂ $b$ ≻₂ $a$ as the following matrix shows:

	a	b	d
b	uni(d,e)	d	uni(a,e)
d	uni(b,e)	uni(a,e)	b
e	uni(b,d)	uni(a,d)	uni(a,b)

$\Gamma \left( {a,b,c,e} \right)$: This game admits a unique equilibrium $\left( {e, a} \right)$ with outcome $uni\left( {b, c} \right)$ as preferences over $\left( {a, b, c, e} \right)$ are given by $a$ ≻₁ $b$ ≻₁ $c$ ≻₁ $e$ and $e$ ≻₂ $c$ ≻₂ $b$ ≻₂ $a$.

	a	b	c
b	uni(c,e)	c	uni(a,e)
c	uni(b,e)	uni(a,e)	b
e	uni(b,c)	uni(a,c)	uni(a,b)

• 4. Profile A2

In this profile $a$ ≻₁ $b$ ≻₁ $c$ ≻₁ $d$ ≻₁ $e$ and $c$ ≻₂ $b$ ≻₂ $a$ ≻₂ $d$ ≻₂ $e$. The matrix of the game is as follows:

	b	a	d	e
b	${\Gamma }\left( {a,c,d,e} \right)$	$c$	a or c	a or c
c	a	b	a or b	a or b
d	a or c	b or c	${\Gamma }\left( {a,b,c,e} \right)$	b
e	a or c	b or c	b	${\Gamma }\left( {a,b,c,d} \right)$

The equilibria for the subgames that with 4 non-vetoed options are determined in the sequel; their outcome is irrelevant to prove that b is the unique equilibrium outcome. For instance, the strategy (c, a) is an equilibrium.

$\Gamma \left( {a,c,d,e} \right)$: This game admits two equilibrium outcomes, $a$ or $c$, since restricted preferences are $a$ ≻₁ $c$ ≻₁ $d$ ≻₁ $e$ and $c$ ≻₂ $a$ ≻₂ $d$ ≻₂ $e$_, as one can deduce from the following matrix.

	a	d	e
c	d	a	a
d	c	a or c	uni(a,c)
e	c	uni(a,c)	a or c

$\Gamma \left( {a,b,c,e} \right)$: This game admits $\left( {c, a} \right)$ which leads to $b$ as the unique equilibrium as implied by the preferences $a$ ≻₁ $b$ ≻₁ $c$ ≻₁ $e$ and $c$ ≻₂ $b$ ≻₂ $a$ ≻₂ $e$, as the matrix shows.

	a	b	e
b	c	a or c	uni(a,c)
c	b	a	uni(a,b)
e	uni(b,c)	uni(a,c)	b

$\Gamma \left( {a,b,c,d} \right)$: This game admits $\left( {c, a} \right)$ with outcome $b$ as the unique equilibrium as implied by preferences $a$ ≻₁ $b$ ≻₁ $c$ ≻₁ $d$ and $c$ ≻₂ $b$ ≻₂ $a$ ≻₂ $d$.

	a	b	d
b	c	a or c	uni(a,c)
c	b	a	uni(a,b)
d	uni(b,c)	uni(a,c)	b

• 5. Profile A3

In this profile $a$ ≻₁ $b$ ≻₁ $c$ ≻₁ $d$ ≻₁ $e$ and $b$ ≻₂ $a$ ≻₂ $c$ ≻₂ $d$ ≻₂ $e$. The matrix of the game is as follows:

	a	c	d	e
b	${\Gamma }\left( {c,d,e} \right)$	${\Gamma }\left( {a,d,e} \right)$	${\Gamma }\left( {a,c,e} \right)$	${\Gamma }\left( {a,c,d} \right)$
c	${\Gamma }\left( {b,d,e} \right)$	${\Gamma }\left( {a,b,d,e} \right)$	${\Gamma }\left( {a,b,e} \right)$	${\Gamma }\left( {a,b,d} \right)$
d	${\Gamma }\left( {b,c,e} \right)$	${\Gamma }\left( {a,b,e} \right)$	${\Gamma }\left( {a,b,c,e} \right)$	${\Gamma }\left( {a,b,c} \right)$
e	${\Gamma }\left( {b,c,d} \right)$	${\Gamma }\left( {a,b,d} \right)$	${\Gamma }\left( {a,b,c} \right)$	${\Gamma }\left( {a,b,c,d} \right)$

where $\Gamma (X$) stands for the subgame where players bargain over $X$. Whenever $X$ has three options, the outcome is determined directly by the analysis in Section A.2. This directly implies that the payoff matrix can be simplified as follows:

	a	c	d	e
b	c	a*	a	$a$
c	b*	${\Gamma }\left( {a,b,d,e} \right)$	a or b	a or b
d	b	$a{\text{ or }}b$	${\Gamma }\left( {a,b,c,e} \right)$	a or b
e	b	$a{\text{ or }}b$	$a{\text{ or }}b$	${\Gamma }\left( {a,b,c,d} \right)$

The subgames that involve 4 options are determined in the sequel. Each of these subgames admits an equilibrium; moreover, their equilibrium outcome is irrelevant to determine the set of equilibrium outcomes over the whole game. It follows that the equilibrium outcomes are $a$ (sustained by the profile $\left( {b, c} \right)$) and $b$ (sustained by the profile $\left( {c, a} \right)$), as wanted.

$\Gamma \left( {a,b,d,e} \right)$: This game admits two equilibrium outcomes, $a$ and $b$, with the restricted preferences $a$ ≻₁ $b$ ≻₁ $d$ ≻₁ $e$ and $b$ ≻₂ $a$ ≻₂ $d$ ≻₂ $e$. The payoff matrix is as follows.

	a	d	e
b	d	a	a
d	b	a or b	uni(a,b)
e	b	uni(a,b)	a or b

$\Gamma \left( {a,b,c,e} \right)$: This game admits two equilibrium outcomes, a or b, with the restricted preferences $a$ ≻₁ $b$ ≻₁ $c$ ≻₁ $e$ and $b$ ≻₂ $a$ ≻₂ $c$ ≻₂ $e$.

	a	c	e
b	c	a	a
c	b	a or b	uni(a,b)
e	b	uni(a,b)	a or b

$\Gamma \left( {a,b,c,d} \right)$: This game admits two equilibrium outcomes, $a$ or $b$, with the restricted preferences $a$ ≻₁ $b$ ≻₁ $c$ ≻₁ $d$ and $b$ ≻₂ $a$ ≻₂ $c$ ≻₂ $d$.

	a	c	d
b	c	a	a
c	b	a or b	uni(a,b)
d	b	uni(a,b)	a or b

• 6. Profile A5

In this profile a ≻_i b ≻_i c ≻_i d ≻_i e for $i$ = 1, 2. The restriction that players rule out weakly dominated strategies implies that neither of the players vetoes $a$ at any stage. Thus, the only outcome of the game is $a$ since in any subgame $a$ is always unanimously preferred and hence never vetoed.

B. Coordination Patterns in Experimental Sessions

In each of the experiments analyzed in this paper, the subjects play together for a certain number of rounds. It is thus possible that they learn how to play a strategy that increases their payoff during the experiment. Consequently, the results of games played in the first rounds of the experiment may differ from the ones in the last rounds. The tables below report the results of the experimental session by distinguishing the games happening at the beginning of the sessions and those at the end. In doing so, we aim to identify possible coordination and learning patterns between subjects.

3.1 B.1. Coordination Patterns with the Non-Symmetric Shortlisting Mechanism

In the data of De Clippel et al. (2014), there are 40 rounds in each session. To identify coordination patterns between subjects, we reproduce Tables 3 and 4 of the main text by splitting the sample in two: the games played in the 20 first rounds of the session on the one hand and those played in the 20 last rounds on the other. In our data, there are also 40 rounds per session, so we use the same split-sample strategy. Note however that the preference profile A0 was always played first. so there is an empty cell for this profile in the rows for the 20 last rounds.

We find some coordination patterns between subjects, but only a few are systematic across preference profiles. The patterns are indeed quite different depending on the profile: in some, efficiency increases and inequality decreases as the experiment advances (e.g. Pf5), in others, this pattern is reversed (e.g., A2). The most systematic pattern deals with the first-mover advantage: the first-mover advantage tends to increase as the experiment progresses in both De Clippel et al. (2014) and our data. This pattern is observed in virtually all profiles. It suggests that subjects learn how to take advantage of their position during the experiment.

	Efficiency		Inequality
	20 first rounds	20 last rounds	20 first rounds	20 last rounds
A0	20.00	20.00	3.00	1.90
A2	27.96	27.55	5.92	6.35
A3	34.62	34.25	4.87	4.83
Pf5	25.40	26.56	8.15	7.56

1. Coordination efficiency and inequality in De Clippel et al. (2014).

	First-mover advantage
	20 first rounds	20 last rounds
A0	− 1.17	− 1.20
A2	2.92	4.05
A3	1.72	3.62
Pf5	0.60	5.27

1. First-mover advantage in De Clippel et al. (2014).

	Efficiency		Inequality
	20 first rounds	20 last rounds	20 first rounds	20 last rounds
A0	20.00		2.50
A2	29.00	26.75	2.87	2.75
A3	30.25	30.55	3.65	3.89
A5	38.00	38.05	0.00	0.00

1. Efficiency and inequality in our data (20 first vs. 20 last rounds).

	First-mover advantage
	20 first rounds	20 last rounds
A0	− 0.43
A2	− 1.62	0.45
A3	0.15	1.39
A5	0.00	0.00

1. First-mover advantage in our data (20 first vs. 20 last rounds).

3.2 B.2. Coordination Patterns with Alternative Mechanisms

We reproduce the results of Table 5 of the text using the same sample split-sample strategy as above. Again, we do not find much pattern of coordination between subjects as the experimental session advance. The only (mild) pattern that we find is for the simultaneous mechanism: efficiency and but also inequality increase as the experiment progresses. This suggests that subjects managed to coordinate and learn how to play the most profitable strategy for them as the experiment advanced. Note however that these increases are small, and cannot be found for the gradual vetoes mechanism.

	Efficiency		Inequality
	20 first rounds	20 last rounds	20 first rounds	20 last rounds
A0	20.00		3.07
A2	27.40	28.61	3.70	3.50
A3	27.37	28.77	3.00	3.32
A5	35.50	35.00	0.00	0.00

1. Efficiency and inequality with the simultaneous mechanism (20 first vs. 20 last rounds).

	Efficiency		Inequality
	20 first rounds	20 last rounds	20 first rounds	20 last rounds
A0	20.00		3.04
A2	26.78	27.11	0.78	1.55
A3	31.72	28.75	4.17	3.30
A5	38.89	39.83	0.00	0.00

1. Efficiency and inequality with the gradual vetoes mechanism (20 first vs. 20 last rounds).

3.3 B.3. Full Results

The table below presents the full results of our experimental sessions. It gives the selection rate of each option, $a$, $b$, $c$,$d$, and $e$, under each profile of preferences and mechanism.

	Shorlisting (%)	Simultaneous (%)	Gradual Vetoes (%)
A0
a	2.50	5.36	4.44
b	5.36	9.29	10.74
c	81.79	76.43	75.93
d	6.07	7.14	7.04
e	4.29	1.79	1.85
A2
a	13.21	17.86	5.56
b	60.00	56.07	74.44
c	14.64	17.86	7.41
d	10.36	6. 43	7.78
e	1.79	1.79	4.81
A3
a	47.50	30.71	32.96
b	28.57	33.93	38.89
c	15.71	25.00	20.00
d	6.79	7.50	5.93
e	1.43	2.86	2.22
A5
a	90.71	70.71	96.30
b	2.14	15.71	2.59
c	5.00	9.64	1.11
d	1.07	2.50	0.00
e	1.07	1.43	0.00

1. Rates at which each option is selected in each profile and each mechanism in our data.

C. First-Mover Advantage in Alternative Mechanisms

The table below presents the first-mover advantage for the alternative mechanisms. Unsurprisingly, we do not find any effect given that, in these mechanisms, which subject play “first" or “second" is purely artificial. They indeed both play at the same time. Yet, for coding purposes, there is a first and a second-mover in our Z-tree program. This is what we use in the analysis below.

	Simultaneous	Gradual Vetoes
A0	0.93	0.89
A2	0.00	− 0.19
A3	− 0.16	− 0.30
A5	0.00	0.00

1. First-mover advantage in alternative mechanisms. Entries are average differences between the payoff of the first-mover and the payoff of the second-mover in our data.

D. Analysis of Data from the CREST Lab

For our data, we conducted 6 experimental sessions: 3 in the PSE lab and 6 in the CREST lab. To discard any possibility to our results are driven by a lab effect, we reproduce Table 5 by focusing on data coming from sessions organized in the CREST lab. All three mechanisms were covered in the sessions of the CREST lab whereas only the non-symmetric shortlisting mechanism was covered in the PSE lab. Focusing on data from the CREST lab, we still have 400 games for the simultaneous mechanism (1 session), 800 games for the simultaneous mechanism (2 sessions), and 1080 games for the gradual vetoes mechanism (3 sessions).

	Shorlisting		Simultaneous		Gradual
	Efficiency	Inequality	Efficiency	Inequality	Efficiency	Inequality
A0	20.00	2.60	20.00 (.)	2.95 (0.62)	20.00 (.) (.)	3.04 (0.52) (0.87)
A2	26.10	1.30	27.75 (0.08)	3.15 (0.00)	27.00 (0.36) (0.29)	1.30 (0.99) (0.00)
A3	28.80	3.40	28.77 (0.98)	3.32 (0.8)	29.74 (0.38) (0.26)	3.59 (0.47) (0.21)
A5	38.00	0.00	35.25 (0.00)	0.00 (.)	39.52 (0.00) (0.00)	0.00 (.) (.)

1. Entries are means using CREST experimental data only. In parentheses are p-values of t-tests testing the equality of means between shortlisting versus simultaneous/gradual (first row of parentheses) and between simultaneous versus gradual (second row of parentheses) (two-tailed). N per cell = 100 (shortlisting), 200 (simultaneous), 270 (gradual).

The results are going in the same direction as those that include data from both labs: the results of the gradual vetoes mechanism are not statistically significantly different from those of the non-symmetric shortlisting mechanism, and when they are it is the gradual vetoes mechanism that performs better (i.e. profile A5). By contrast, the performance of the simultaneous mechanism is generally poorer than the non-symmetric shortlisting and gradual vetoes mechanisms (e.g., profile A5) although differences are not always statistically significant due to the relatively low number of observations.

E. Regression Analysis

In our experiment, subjects are grouped by sessions. To account for the non-independence among the sessions and their effects on p-values, we reproduce Table 5 using an OLS regression framework, in which we cluster standard errors by session. The regression model is the following:

$$ Y_{ij} = {\text{A}} + {\text{B}}\beta_{ij} + {\text{C}}\gamma_{ij} + \epsilon_{ij} $$

(3)

where Y is efficiency (first table) or inequality (second table) of each pair of subjects $i$ at session $j$, $\beta$ is a dummy variable capturing the simultaneous mechanism and $\gamma$ a dummy variable capturing the gradual vetoes mechanism. The non-symmetric shortlisting mechanism is thus the reference category. To calculate the p-value of the difference between the simultaneous mechanism and the one with gradual vetoes, we estimate an extra OLS regression:

$$ Y_{ij} = {\text{A}} + {\text{B}}\alpha_{ij} + {\text{C}}\gamma_{ij} + \varepsilon_{ij} $$

(4)

where α is a dummy variable capturing the non-symmetric shortlisting mechanism. The reference category is thus the simultaneous mechanism. We find that results hold: differences in efficiency are statistically significant in particular for preference profiles A3 and A5, and differences in inequalities are statistically significant in profiles A2 and A3.

	Profile A0		Profile A2		Profile A3		Profile A5
	(1)	(2)	(1)	(2)	(1)	(2)	(1)	(2)
Shorlisting	(ref)	(.)	(ref)	− 0.79 (0.82)	(ref)	2.07* (1.01)	(ref)	2.86*** (0.85)
Simultaneous	(.)	(.)	− 0.79 (0.82)	(ref)	− 2.07* (1.02)	(ref)	− 2.86*** (0.85)	(ref)
Gradual	(.)	(.)	− 0.40 (1.05)	− 1.18 (0.90)	− 0.71 (0.94)	1.48 (0.98)	1.45 (0.88)	4.34*** (0.32)

1. Entries are coefficients from OLS regressions predicting efficiency. In regression (1), the non-symmetric shortlisting mechanism is the reference category, in regression (2) the simultaneous mechanism is the reference category. In parentheses are standard errors clustered by session. *p < 0.1, **p < 0.05, ***p < 0.01.

	Profile A0		Profile A2		Profile A3		Profile A5
	(1)	(2)	(1)	(2)	(1)	(2)	(1)	(2)
Shortlisting	(ref)	− 0.57 (0.58)	(ref)	− 0.79 (0.92)	(ref)	0.57* (0.30)	(ref)	(.)
Simultaneous	0.57 (0.58)	(ref)	0.79 (0.92)	(ref)	− 0.57* (0.3)	(ref)	(.)	(ref)
Gradual	0.53 (0.64)	− 0.03 (0.65)	− 1.49 (0.82)	− 2.28*** (0.56)	− 0.21 (0.38)	0.36 (0.3)	(.)	(.)

1. Entries are coefficients from OLS regressions predicting inequality. In regression (1), the non-symmetric shortlisting mechanism is the reference category, in regression (2) the simultaneous mechanism is the reference category. In parentheses are standard errors clustered by session. *p < 0.1, **p < 0.05, ***p < 0.01.

F. Analysis of Individual Decisions (Same Vetoes)

The table below presents the detail of individual decisions of subjects under the gradual vetoes mechanism. It shows the rate of common vetoes, i.e., vetoes that are put on the same option by both subjects at the same time, at each round of elimination: first, second, and third. Mechanically, it takes at least two rounds to reach a final selected option. 63% of all games finish there, and 96% finish after a third round. Since the numbers are too small, we do not include the fourth round in the table (depending on the profile, the N is as small as 8). We observe that the rate of common vetoes increases with the number of rounds. This is not particularly surprising given that there are fewer and fewer options available and that subjects are likely to pick the same veto, even by chance.

	Same Vetoes
	Round 1 (%)	Round 2 (%)	Round 3 (%)
A0	10	7	20
A2	17	33	46
A3	21	39	58
A5	23	30	90

1. Entries are the rate of common vetoes at each round of elimination in the gradual vetoes mechanism.

G. Analysis of Individual Decisions (Other Strategies)

To further study the individual decisions made by subjects during the experiment, we define two intuitive veto strategies: a cautious veto and an aggressive veto. The cautious strategy consists in putting a veto on the option associated with the smallest personal payoff for the subject among the remaining options. The aggressive strategy consists in putting a veto on the option associated with the largest payoff for the opponent. The first table below reproduces Table 8 of the main text and presents the share of cautious and aggressive vetoes adopted by subjects. To make the results comparable across mechanisms, we focus on the two first vetoes put by each subject under both mechanisms. In the case of the simultaneous mechanism, this means analyzing the universe of decisions made by subjects. For the gradual vetoes mechanism, this means focusing on the two first rounds of elimination. From the first table below, we observe both strategies capture a substantial share of individual decisions. We also observe that while the share of cautious vetoes tends to be larger under the simultaneous mechanism compared to the one with gradual vetoes, the share of aggressive vetoes tends to be lower, regardless of the preference profiles. Note however that overall the individual strategies seem to follow similar patterns under both mechanisms in the sense that the share of cautious and aggressive vetoes is mostly a function of the preference profile. For example, nobody has adopted an aggressive strategy in profile A5, which is not a surprise given that both paired subjects share the same preference for the options available.

To further explore individual decisions, we construct a second table in which we report the share aggressive and cautious vetoes under the gradual vetoes mechanism at each round of elimination: first, second, and third. The most striking pattern is that, with the exception of Profile A0, the rate of cautious vetoes increases with the round of elimination, whereas the rate of aggressive vetoes decreases (at least from the first to the second elimination round). This suggests that subjects adopt a strategy of first acting aggressively, probably as they anticipate that their opponent will seek the options associated with their highest payoff, and then, as the final decision is getting closer, they secure their payoff by removing the option associated with their smallest personal payoff.

	Simultaneous		Gradual
	Cautious	Aggressive	Cautious	Aggressive
A0	86%	93%	82% (0.01)	94% (0.15)
A2	52%	34%	43% (0.00)	42% (0.00)
A3	58%	18%	50% (0.00)	20% (0.26)
A5	60%	0%	58% (0.31)	0% (.)

1. Entries are proportions of vetoes put on the option associated with the smallest personal payoff of the subject (cautious) and on the option associated with the largest payoff of the opponent (aggressive) among the remaining options (out of two vetoes). In parentheses are p-values of t-tests testing the equality of proportions between simultaneous versus gradual. N per cell = 1120 (simultaneous), 1080 (gradual).

	Cautious			Aggressive
	Round 1 (%)	Round 2 (%)	Round 3 (%)	Round 1 (%)	Round 2 (%)	Round 3 (%)
A0	72	94	85	93	96	88
A2	31	67	95	53	32	53
A3	38	72	88	24	16	31
A5	54	69	92	0	0	0

1. Entries are the rate of cautious and gradual vetoes at each round of elimination in the gradual vetoes mechanism.