Bargaining and Coalition Formation Dr James Tremewan - - PowerPoint PPT Presentation

Bargaining and Coalition Formation

Dr James Tremewan (james.tremewan@univie.ac.at) Experiments on Cooperative Solution Concepts

Experiments on Cooperative Solution Concepts

Experiments on Cooperative Solution Concepts

• ”A test of the core, bargaining set, kernel and Shapley models in

N-Person Quota Games with one weak player,” Horowitz (1977).

• Experiments on four and five-player versions of the ”Three

Player Game” we looked at last week.

• ”An Experiment on a Core Controversy,” Yan and Friedman

(2010).

• Experiments on the ”Glove Market” from last week, and a five

player version.

• ”Committee decisions under Majority Rule: An Experimental

Study,” Fiorina and Plott (1978).

• Classic paper testing a variety of theories in ”spatial games.”
• ”Patterns of Distribution in Spatial Games,” Eavey (1991).
• Challenges some of the conclusions of the previous paper.

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Horowitz (1977)

Horowitz (1977)

• Experiments on ”n-person (n-1) quota games” with one weak

player for n ∈ {4, 5}.

• n-player game where the value of a coalition is positive if and
• nly if it contains at least n-1 players.
• Values of coalitions are chosen such that in core allocations:
• One player (the weak player) will receive zero.
• A different range of payoffs is predicted for the other players.
• The value of the grand coalition is equal to the highest value of

(n-1) coalitions.

• Example for n = 3 from last week:
• v({1, 2, 3}) = 10,
• v({1, 2}) = 10, v({1, 3}) = 3, v({2, 3}) = 2,
• v({1}) = v({2}) = v({3}) = 0.

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Horowitz (1977)

3-Player Quota Game With One Weak Player

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✉ ✉ ✉

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(0,10,0) (10,0,0) (0,0,10) (8,2,0) (3,7,0) Shapley value Core

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Horowitz (1977)

Horowitz (1977): Implementation

• Mediated through computers.
• Comunication occured in fixed order, and restricted to six

keywords: offer, accept, reject, ratify, pass, solo.1

• If an offer was accepted by enough people, there would be a

ratification stage where a preliminary agreement becomes binding.

• Results: as predicted by the core, the weak player always

received zero!

1”Solo” meant withdraw from game which is pointless in these

games... the program (coalitions: see Kahan and Rapaport, 1974) was designed for general games in characteristic form.

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Horowitz (1977)

Horowitz (1977): Results (4 Players)

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Horowitz (1977)

Horowitz (1977): Results (5 Players)

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Horowitz (1977)

Horowitz (1977): Results

• The core does a good job of predicting outcomes:
• The weak player always gets zero.
• For 8/9 types of players, the average payoff was in the core.
• 90/108 individual payoffs were within the core.
• Where payoffs fell outside of the core they were in the direction
• f the Shapley value.

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Yan and Friedman (2010)

Yan and Friedman (2010)

• Market Game:
• s sellers with cost c and b buyers with valuation v, where

v = c + 1.

• Each seller has precisely one good, and each buyer can buy one.
• For s = 1 and b = 2 this game is equivalent to the Glove Market

game we looked at last week.

• If s > b the sellers are on the ”long side” of the market and

buyers on the ”short side” (and vice versa).

• In general, the core predicts that the players on the short side

will extract all the surplus.

• This results in the extreme prediction that switching one buyer

to a seller can switch the entire surplus from one side to the

• ther, often seen as a bad feature of the core: intuitively players
• n the long side should be able to collude in some way.
• This paper tests this prediction for s = 1 and b = 2, and s = 2

and b = 3.

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Yan and Friedman (2010)

Yan and Friedman (2010): Implementation

• Uses continuous double auction (CDA): Buyers and seller post

bids and offers publicly, which can be accepted in real time.

• Three treatments:
• Std: no communication (apart from bids and offers).
• Chat: free pre-play (public) communication in online chatroom.
• Barg: allows chat, and for players on the long side to form

collusive agreements.

• A collusive agreement means one or more players withdraw from

trading and give another the right to buy or sell their good in exchange for a proportion of the profit. These agreements can be withdrawn from unilaterally, but only before a trade has been implemented.

• Designed to give collusion its best shot.
• After 8 periods, a “long-side” subject switches roles.
• In following slides, SSS (Short Sider Surplus Shares) = the

fraction of the surplus obtained by the short sider.

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Yan and Friedman (2010)

Yan and Friedman (2010): Results

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Yan and Friedman (2010)

Yan and Friedman (2010): Results

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Yan and Friedman (2010)

Yan and Friedman (2010): Results

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Yan and Friedman (2010)

Yan and Friedman (2010): Results

• Median share obtained by short side player (1x2 games, similar

results for 2x3):

• Std: 0.912
• Chat: 0.706
• Barg: 0.735
• Paper claims this is strong evidence in favour of the core

prediction as observations are significantly closer to this (1) than the equal split (0.333).

• Certainly, switching one player from buyer to seller has a large

impact on outcomes as predicted by the core. (Would be more spectacular with bigger markets, but expensive).

• Comment: It is not immediately obvious why this is the

appropriate comparison to make, for example observations in the last two treatments are close to the Shapley value (0.666). Ideas

• f fairness may take into account strategic advantage.

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Fiorina and Plott (1978)

Fiorina and Plott (1978)

• Spatial Games:
• Players asked to select as a group a point on a plane (i.e. the

blackboard).

• Each player has a different preference ordering over the points:

essentially an ”ideal point” which results in their highest payoff, with payoffs for other points decreasing with distance from that point (private information).

• Players communicate freely face-to-face, but:
• May not make side deals.
• May not discuss their monetary payoff for different points.
• Procedure:
• Initial point (motion on the floor) is the top right corner.
• Any player may propose a new point: if this passes a majority

vote it becomes the new motion.

• At any time there can be a vote on a motion to end debate,

which if passed ends the experiment.

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Fiorina and Plott (1978) 16/28

Fiorina and Plott (1978)

Fiorina and Plott (1978): Results and Conclusions

• Sessions were run for high and low incentives, and three different

preference configurations.

• 16 different models were considered, from economics, sociology,

political science etc.

• All models performed poorly in the low incentives sessions,

where outcomes tend to be more central (fairer).

• In the high incentives treatment the outcomes were tightly

clustered around the core (in the two configurations where it existed).

• Note that the core coincided with the van

Neumann-Morgernstern solution and the ”Voting Equilibruim” (Plott and Sloss, ?).

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Eavey (1991)

Eavey (1991)

• In the previous paper (FP) there were no points where all players

got a substantial payoff, i.e. a point satisfying the ”something for everyone” fairness norm.

• Eavey argues that the core performed well in FP because there

were no ”fair” alternatives.

• This paper uses the same design as FP (as well as Hoffmann and

Plott (1983) who just change the procedural rules), but alter the payoff functions.2

• Note that the preference orderings are the same, so the core

does not change.

• The payoff functions are such that there is now a region (closer

to player 5 than the core) where all players get a substantial payoff.

2Two other games are also studied which we will not discuss here.

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Eavey (1991)

Eavey (new) vs Fiorina and Plott (original)

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Eavey (1991)

Eavey (new) vs Hoffmann and Plott (original)

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Eavey (1991) 21/28

Eavey (1991)

Eavey (1991)

• Conclusion: when fair outcomes are available, outcomes drift

away from the core to fairer alternatives.

• BUT! Maybe this is just weak incentives!:
• Note the frequency of round numbers in Eavey’s results

compared to the original studies (previous slide). This could indicate ”satisficing” behaviour: the incentives are flat so there is no point wasting too much effort getting it just right.

• See the next slides for a comparison of the two incentive

schemes (much flatter for Players 1,2, and 3, slightly steeper for 4, and the same for 5).

• FP had already noted that fairness concerns seemed only to

play a role when size of incentives were substantial.

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Eavey (1991) 23/28

Eavey (1991) 24/28

Eavey (1991) 25/28

Eavey (1991) 26/28

Eavey (1991) 27/28

Eavey (1991)

General Conclusions

• The core generally predicts experimental outcomes well.
• However, for many games the core does not exist, or is too large

to provide a useful prediction.

• Sometimes there is a tendency towards ”fairer” allocations like

the Shapley value, especially where incentives are weak (e.g. low incentive treatment in FP, and possibly Eavey).

• Fairer outcomes may also be more prominent where ordinal

payoff information is public, so interpersonal comparisons are

• facilitated. This was specifically prevented in FP.

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