Verified by 
Just say no.
Just say no.
~ Nancy Reagan
In the Roman Republic, the people’s tribune had the power to frustrate the aristocrats by saying veto (“I forbid it”) to proposed legislation. The veto is a powerful strategy when it takes two parties to agree to a plan. He who has veto power is stronger than he who does not, but it is a negative power. It contributes to new endeavors only by tolerating them.
In Power Games, I began to explore Steve Laffey’s dictum that “there’s tremendous power in not giving a crap” from a game-theoretic perspective. I showed that the Ultimatum Game captures this type of power, whereas the Dictator Game, the Game of Chicken, and the Prisoner’s Dilemma do not. I looked at these games assuming that there is only one round of play. Today, I go farther with Steve Brams’s (e.g., 2011) Theory of Moves (ToM)
In ToM, as in other forms of game theory, we have two players, Row and Column, each with two strategies to choose from. The strategies may be cooperation and defection, but other labels could be more appropriate depending on the specifics of the game. I use the labels Play, P, and Rest, R, where play refers to up and left respectively for Row and Column, and rest respectively refers to down and right. Why these labels? I want to explore Column’s veto power. Column knows Row’s preferences. Row would love to play with Column (PP = 4), but it takes two to tango (have sex, eat dinner, play chess, or whatever). If mutual play cannot be had, Row prefers to rest with Column (RR = 3). Row would rather not rest if Column wants to play (RP = 2), and he would really hate to propose play and be rejected (PR = 1). Row’s payoff ordering is thus PP > RR > RP > PR, and Column knows it.
Column’s veto power, her power to make Row miserable, depends on her own payoff ordering. If her preferences were the same as Row’s, there would be no power differential; Row and Column would play happily ever after. But we know this to be a fairy tale; so let’s move to reality.
I will – for now – assume that Column also likes mutual play, but not as much as Row does. She likes it enough for Row to be drawn to her, but not so much that she has no veto power. For her, PP is 3. The other three payoffs can vary to create 6 distinct combinations. What do these 6 games entail for the relationship between Mr. Row and Ms. Column?
Game 1
Column
P R
Row P 4,3 1,2
R 2,1 3,4
Column (Col) wants mutual rest the most, and would least likely initiate play when Row is not even up for it. In a ToM analysis, we assume that both preference rankings are common knowledge (i.e., known to both players), but we need to decide who goes first. Then we can consider the possibility of countermoves, perhaps ad infinitum – or at least ad nauseam. If Row goes first, he will choose P, knowing that P is Col’s best response. If Col goes first, she will choose R for the same reason. PP and RR are equilibria because neither player can do better by changing strategy unilaterally. This seems to say that Col has no veto power over Row. But – and this is a unique feature of ToM – players may be strategically looking past the next move. After PP, Col may go again and shift to R, knowing that Row will R in return, which results in RR, which is good for Col. Col’s veto power lies in letting Row go first. If she goes first, however, Row has veto power over her. Overall, the game is symmetrical, as the payoff matrix shows.
Game 2
Column
P R
Row P 4,3 1,1
R 2,4 3,2
Col’s dominating strategy is to P. Knowing this, Row will P and get what he wants. To get what she wants, Col needs to R, knowing that Row will also R, in response to which Col can P. RP is not an equilibrium, though, and Row will P again. If Col has the guts to take the duo through two terrible states, PR and RR, to temporarily get RP, the cycle can go on forever, leaving both players with the dissatisfying average payoff of 2.5. This is merely a shadow of veto power.
Game 3
Column
P R
Row P 4,3 1,4
R 2,1 3,2
We now assume that R is Col’s dominating strategy. Row needs to R to secure his second best payoff. If he chose P, he would have to anticipate Col’s veto. The veto, in other words, will not even be exercised (if Row is rational). Its power lies in its threat. If Col is dissatisfied with having her second worst payoff, she can P, knowing that Row with P as well, in which case she can R (now exercising her veto) and settling on a cycle. For Col, cycling is better than the equilibrium of RR (2.5 > 2), but for Row it is not. Therefore, if Col is clever, she will force the cycle, which Row is unable to stop. This is not really veto power, but cycle power.
Game 4
Column
P R
Row P 4,3 1,1
R 2,2 3,4
This game is like game 1, except that Col’s two lowest preferences are reversed. It is also like game 3 because the players either accept an unequal equilibrium or cycle forever and benefit less. Again, there is no strong case for Col’s veto power.
Game 5
Column
P R
Row P 4,3 1,2
R 2,4 3,1
This game is like game 2, again with Col’s two lowest preferences reversed. Col’s dominating strategy is P, and she can veto Row’s P only by inducing a cycle to both player’s loss.
Game 6
Column
P R
Row P 4,3 1,4
R 2,1 3,2
Finally, we have a game, which is like game 3 with Col’s lowest preferences re-ordered. Row’s best first move is again to R in order to get his second best outcome. Col can again force a cycle to attain a small improvement in her average payoff.
All told, it is not possible to give Col strong veto power over Row if PP is Col’s second preference. Col can initiate cycles to get her best outcome on every fourth round but her average payoff is less than it would be if she accepted mutual play in games 1, 2, 4, and 5. In games 3 and 6, her cycling gives her a small advantage over the equilibrium of RR, which she does not like very much.
Game 7
Column
P R
Row P 4,1 1,2
R 2,3 3,4
There is a game that would give Col more power. In this game, her dominating strategy is R, and Row will R because he knows it. If PP were ever to arise – because both players were drunk perhaps, Col would soon veto it by going to R and Row would follow suit. Row cannot force a cycle out of this equilibrium. Col wins, probably without ever having to exercise her veto power. The question is why they ever started interacting if they have no rational way of getting to play together, even if only temporarily.
In the games I have considered, veto power affects both the veto-er and the veto-ee. 2 x 2 games involve interdependence, unless a person does not care what the other does. There is no way for one to have power over the other without herself also being the subject of the other's moves. In interdependent games it is pointless to look for a pure and total asymmetry of power. It is also pointless trying to separate personal from social power. Lammers, Stoker, & Stapel (2009) suggested that personal power rests in the ability to maintain independence from another's influence attempts. Social power, in contrast, is the ability to control others. Lammers et al. show that these types of power can be separated theoretically and psychologically. Behaviorally, however, one constitutes the other, until you empower yourself by choosing not to play.
Brams, S. J. (2011). Game theory and the humanities. Cambridge, MA: MIT Press.
Lammers, J., Stoker, J. I., & Stapel, D. A. (2009). Differentiating social and personal power. Psychological Science, 20, 1543-1549.
