More options can lead to worse outcomes
These seem intuitive:
(1) Adding to your options leads to improved outcomes, or at least cannot lead to worse outcomes.
(1') Eliminating options leads to worse outcomes, or at least cannot lead to better outcomes.
The reasoning seems straightforward: If our only option was A, but now option B is added, we could always ignore option B, leaving us as well off as before.
The converse seems straightforward as well. If we had options A and B, and B is now eliminated, then we would not expect that we could be better off. Either A is better than B, in which case B was of no benefit and eliminating it leaves us equally well-off. Or B was better than A, in which case we are now worse off because of the elimination of the better option.
It turns out that (1) and (1') are true only in a narrow sense, and not in most real world situations. Adding options while keeping everything else constant cannot lead to worse outcomes, while removing options while keeping everything else constant cannot lead to better outcomes. Among other things, keeping everything else constant means that nobody knows about the option except the person who has it. Because changes in options usually arise from the choices of others, and/or changes in the material world, they will usually be known by others.
It is therefore rarely possible in real-world situations to easily conclude that a person’s increased options do not leave her worse off, or that her decreased options do not leave her better off. To reach this conclusion, we would either need to know that everything else in the world remains equal, or we must analyze how the other changes impact the individual whose options have changed.
Bracket Hamlet
When I talk about more options or fewer options changing payoffs, I am not primarily talking about the fact that people can have negative psychological reactions to increased options, such as the stress of having to make choices. This is a real factor, but it is not my focus because it is an intuitive dynamic, and one about which not much needs to be said. The stress of making choices is basically friction, but it could be significant — see Hamlet — and could outweigh the advantage of having the extra choice. Let’s not ignore this when doing political and economic analysis, but let’s set it aside so we can get to the main point.
Mechanics and Scenarios
Mechanically, it is not difficult to show that an increase in options can lead to worse outcomes if all else is not equal. Let’s assume a world that includes these entities:
Two people: P and Q
Two options belonging to P: O (the original option) and O’ (the new option).
One observable change in the world that accompanies O’, whatever it may be: Delta.
One rule that person Q follows: when Delta is observed, try to kill P.
Probabilities: there is a 100% chance that Q will observe Delta, and a 100% chance that any murder attempt by Q will succeed.
Two preference orderings of P: P would rather be alive than dead, and the value of P’s life to P exceeds the value of the outcome of O’ minus the outcome of O to P.
In this case, having the additional option O’ makes P worse off, because it is necessarily accompanied by a change in the world that causes Q to kill P, which outweighs any benefit of O’ to P.
While this is formally correct, and sufficient to show that it should not be assumed that more options leave a person not worse off, this is not a particularly realistic scenario. But there are plausible scenarios where an increase in P’s options observed by Q could cause Q, a normal person with normal reasoning and motivations, to act in such a way that leaves P worse off than if P did not have the new option. And conversely, a normal person Q observing a decrease in P’s options could act in a way that improves P’s outcomes. Here are some such scenarios.
Scenario 1: Bridge-burning as credible threat
This scenario is taken from this video. Two countries are at war. Between the countries is an island. Country A’s army has occupied the island, and is considering burning the bridge behind it that would allow the army to retreat back to the country.
In the payoffs stated in the video, it makes sense to burn the bridge in order to limit the army’s options. If Country A burns the bridge, Country B will know that Country A’s army cannot retreat from the island back to Country A, and Country A’s army will therefore put up a fight that is costly to both countries. Since fighting will leave Country B worse off, it would rather acquiesce in Country A’s conquest of the island. If Country A’s army had the option of retreating, Country B would reasonably believe that if it attacked, the enemy army would retreat, leaving Country B better off attacking than acquiescing. In this way, Country A’s limiting of its own options leads to it keeping the island by changing Country B’s calculation.
Scenario 2: Fantasy Football Free Agent Auctions
In fantasy football, it is common for free agents to be acquired in auctions in which bids are secret and can be modified or cancelled. These bids are made in in-game currency known as FAAB. It is a feature of such leagues that a player benefits from an opponent’s depletion of FAAB, since it gives the player a monetary advantage in future auctions. When determining a bid amount, one has to figure out how much FAAB the player is worth, as well as how much FAAB other players are expected to bid.
Suppose I believe that a player is worth 50 FAAB to me but more than 50 FAAB to one or more of my opponents (for example, because I have a better team, so the player represents less of an improvement, or because I have less FAAB and therefore value it more). Suppose that the depletion of one of my opponents’ FAAB is worth more to me than obtaining the player at a good value.
If the auction were a regular (“English”) auction, I could accomplish my goal by bidding 50, thereby forcing my opponents to bid 51. My opponents would know that I am bidding 50 and cannot retract the bid, so in order to obtain the player, they would need to bid 51 or more, to my benefit.
In a secret auction with retractable bids, I cannot force my opponents to bid high, since my opponents do not know my bid and would not necessarily trust that I would not retract the bid. I could announce a bid of 50 and even screenshot it, but my opponents could suspect that it is a bluff designed to motivate them to bid high, and that in fact I would retract the bid and enter a lower one.
In other words, me having the option of retracting my bid means that the bid is not credible to others, which removes their incentive to outbid me, to my detriment.
