Power is the ability to influence the behavior of other agents and could emanate from different sources, for instance political, military, or monetary.
AES’s main stability concept, self-enforcement
, requires that there will be no subcoalition of the winning coalition that will be powerful enough to encourage further deviations. Self-enforcement is a robustness property that ensures that the coalition that forms will never disintegrate thereafter. Another appealing axiom they use is rationality
, which requires agents to pick the coalition that gives them their highest pay-off among self-enforcing coalitions. Rationality is related to immunity to group manipulations in models discussed by Bogomolnaia and Jackson [6
], Ehlers [7
], and Juarez [8
Jandoc and Juarez [9
] extend the analysis of Acemoglu et al. [4
] to the case where power accumulates, as well as to the case when the prize is shared equally among the agents.
Here and in succeeding sections, we use the term dictator if an agent is able to deviate and form a singleton coalition.
We use myopia
as the tendency of agents to choose earlier rewards over future ones. There are proposed explanations for why people choose short-term rewards. First, agents’ time preference may exhibit hyperbolic discounting
where future rewards are more heavily discounted the longer is the horizon (Frederick et al. [10
]). Second, agents may exhibit imperfect forecasting
where agents fail to judge or forecast the value of rewards that occurs further in time (Gabaix and Laibson [11
]). This might happen, for instance, when agents lack the ability to properly backward induct. In this paper, we use the terms “myopic” and “shortsighted” interchangeably, and is meant to describe agents that prefer to form coalitions that may potentially reduce their future payoffs compared to playing the AES equilibrium strategies.
In the AES model, agents outside the winning coalition are killed and the surviving agents can therefore learn from observing past actions of agents that were killed. Indeed, with a large population, such inter-game learning can affect subsequent stages of the coalition formation game. In our experimental bargaining setup with three agents, however, there will be no scope for inter-game learning since the only feasible deviation is into a two-person coalition which will end in the subsequent round either as a dictator or a two-person coalition. Since we play the experiment repeatedly, we will mainly capture the effect of intra-game learning, or learning by playing many subsequent games.
Tremewan and Vanberg [26
] point out that if the outcome from these experiments deviates from theoretical predictions, failure will emanate from behavioral rather than from procedural assumptions.
It is common to assume to restrict the game
such that the power profile
has no ties in the power of any two coalitions; that is,
. Please note that the games that do not satisfy this condition has a Lebesgue measure equal to zero, so this is a weak condition (See Jandoc and Juarez [9
], Acemoglu et al. [4
The pay-off in AES assumes an arbitrarily small cost that is interpreted as the cost of eliminating some of the players from the coalition or as a cost in organizing new coalitions.
While our paper focuses on testing the game proposed by AES, where the determination of the ruling coalition is done sequentially, we note that this assumption is strong and comes with limitations, such as the larger information-exchange needed for the determination of a ruling coalition and applicability to real-world events (e.g., elections or battles in business), where the selection of the ruling coalition is often done simultaneously. A natural extension not studied in our paper is the determination of the equilibrium and its testing for simultaneous versions of the AES’s game.
We note that the determination of the self-enforcing coalition is very robust to the selection of power. In particular, for a three-agent society, unless one agent is a dictator, or two agents have each exactly 50% power, the only possible self-enforcing coalition is the grand coalition.
Please note that this screen is only viewed by those agents included in the proposed coalition. If they are not part of the proposed coalition, their vote is an automatic “No” and they must wait to see whether the proposed coalition passes or fails. If the proposed coalition passes, then they must wait until the game ends.
The players’ payoffs are in real money (US dollars) to make the amount more salient as has been done in the literature on high-stake games, e.g., Cameron [34
We announced the stake prior to the start for each of these final games regardless whether it was a high stake (i.e., $20 or $50) or a low stake (i.e., $5). We implemented the change in stakes in the last round to ensure that agents have learned the game sufficiently and any change in behavior would be purely attributed to the stake.
There are only 3 out of a possible 624 group games in which the grand coalition forms because all agents rejected every proposal made.
We also include a specification where we define the late games to exclude the last game of each session.
However, it is also instructive to examine proposals after a deviation to a two-person coalition if it is part of a person’s empirical best response strategy. This we do in Section 5.2
Please note that there are no singleton coalitions in Table 3
because agents do not have sufficient power to propose themselves at the start of every game.
The complete details on how to obtain the expected payoffs in these tables are shown in Appendix B
There may be other explanations as to why subjects propose or accept two-person coalitions, such as errors (or anticipation of errors) and reciprocity. If there is a small chance of the weaker player accepting a two-person coalition by mistake, then there it is in the best interest of the stronger player to propose such a coalition. On the other hand, a weaker player proposing a two-person coalition can be viewed as a trust game. The weaker player takes the risk in increasing the other player’s share in the hope that the stronger player reciprocates by not eliminating him. These issues are beyond the scope of this paper, and so we leave these for further studies.
This is also consistent with the results in Table 3
where the grand coalition constitutes 60% of Agent 20’s proposals while coalition
only constitutes 6% of his proposals. Agent 35, on the other hand, proposes the grand coalition 20% of the time while proposing
only 7% of the time.
In Appendix C
, we further analyze the proposal and response strategies using a logit model that further corroborates the main findings of this section.
See Nagel and Fang [37
] as well as Selten [38
] for an axiomatic analysis of the method.
Here we note that Agent 20’s empirical best response coincides with his subgame perfect equilibrium strategies.
The only case where it does not happen is in the late games, when Agent 45 proposes and, if passed, about 60% of the time this two-person coalition still forms.
Equal sharing or proportional sharing satisfy a property whereby agents have the same ordinal ranking over coalitions in which they belong. See Jandoc and Juarez [39
] for a more detailed discussion of this property they call “consistent ranking”.
Please note that if the grand coalition is proposed, the coalition always passes, since it is the default starting coalition when a majority of responders votes “no” for it.