The Psychology of Bluffing

If you are going to bluff, make it a big one. —Amarillo Slim

In competitive play, a successful bluff respectively causes delight and consternation in the winner and in the loser—if the bluff is ever revealed. Expectations of the emotional aftermath can co-determine whether a player attempts to bluff and whether the opponent challenges the move. The bluff and the challenge require risk tolerance. A risk-averse person should stay away from games of chance, guts, and strategic interdependence.

The root of the word “to bluff” is the Dutch bluffen, to brag. The bluffer seeks to send the opponent a credible signal that they have a stronger hand or a greater will to aggress than they, in fact, do. So, indeed, they brag, even if in the understated form of keeping the proverbial poker face.

The amount the gambler wagers, literally in poker or metaphorically in other fields of a game, is an ambiguous signal, as bolder gambles can be thought to be associated with stronger hands but also with stronger nerves to risk bluffing.

A lot of psychology in terms of the experience the players have with one another as well as their theatrical acumen and social intelligence come into play, making the situation fantastically complex. The following analysis is not meant to unpack the complexity of the game of poker or the dynamics arising in repeated play (Sklansky, 1989). Instead, we will take a look at the outcome structure of a single strategic encounter, where one party has made an assertive move and where the other party is left wondering if a true threat is at hand or a mere bluff (Ellsberg, 1959).

To gain a foothold for analysis, let us return to the expected emotional payoffs. How will players rank their feelings about the conceivable outcomes? Constructing the payoff matrix shown below, where higher numbers indicate greater pleasure, we place the assertive gambler (the first mover) in the columns, and we assume that this player’s decision to throw down the gauntlet is already made. The opponent (the second mover, or Row) player now faces uncertainty as to whether the Column player, in fact, holds a strong hand (left column) or is bluffing (right column). Within each cell, the Column player’s ranked outcomes are shown to the right of the comma. Row now needs to choose between capitulation (folding, top) and defiance (calling, bottom). Row’s payoffs are shown to the left of the comma.

The Game of Bluff

Source: Courtesy of J. Krueger

The payoffs in this example reflect the assumption that bluffing elicits stronger emotional responses in both players. A successful bluff (top right) delights the bluffer (4) and dejects the opponent (1, if this coup is revealed), while a failed bluff has the opposite effect (1,4, bottom right). If Column makes a legitimately strong play, Row is happier when folding (3) than when calling (2), with the reverse effect on Column.

When a bluff works, that is, if Row folds, its bluffiness may never be revealed. Likewise, if Row folds and the Column’s threat is real, Row may never find out. Only if Row responds assertively as well, by calling, can Row learn anything about Column’s strategy. Row’s payoffs are 2 and 4, respectively, if Column is truly strong or bluffing. Although Row would be better off calling (2) than folding (3) if Column’s threat is legitimate, the sum of payoffs for calling (2 + 4) is greater than the sum of payoffs for folding (3 + 1). Although calling is not even a weakly dominating strategy for Row, the expected value of calling is greater than the expected value of folding, unless Column’s probability of bluffing is less than .25. Bluffing with this probability keeps Row indifferent between folding and calling, that is, bluffing with this probability is Column’s equilibrium strategy.

Row’s equilibrium strategy is to call with a probability of 5. Here, the expected values of a legitimate threat and a bluff are the same for Column. So, Column has no incentive to bluff with the low probability of .25, other than the goal of keeping Row indifferent between folding and calling and to make sure that the total expected value of the game is the same for the two players. If at least one of the players plays the equilibrium strategy, neither can have an advantage over the other.

Arguably, Column might prefer Row to fold because Column’s available payoffs are 2 and 4, whereas the payoffs after Row’s calling are 3 and 1. If Column bluffed less—and if Row knew it—Row would fold more. At the limit, Column would play assertively only when having a legitimately strong hand, and Row would fold. We can see in the matrix, however, that this outcome is the second worst for Column, so back we go to a probability of .25 of bluffing.

Aside from the rational probability calculus, which is difficult to execute, Row may prefer to call because calling ensures that the very worst outcome, a successful bluff, cannot occur. This would seem to be a reasonable loss-averse strategy. However, Column might anticipate this heuristic decision rule and bluff more confidently. Row, in turn, might anticipate this and do the math after all. Anticipating this, Column settles for the equilibrium probability of bluffing. Playing with the equilibrium probability is the only way to avoid going down the rabbit hole of trying to predict the opponent’s prediction or one’s prediction (Grüning & Krueger, 2021).

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Again, this is a structural analysis that strips away social intelligence and experience. Ellsberg’s paper on the political uses of madness remains a classic example of a trenchant reconstruction of historical events. Hitler, Ellsberg showed, bluffed his way into the Rhineland in 1936, projecting both unpredictability and a mad willingness to accept total disaster. With the benefit of hindsight, we wish France had called Hitler’s bluff, but this happens after any bluff that was not called at the time.

The person in the street is left with a dilemma. They may want to learn how to bluff more effectively for their own benefit, while also wishing to learn how to protect themselves more effectively from bluffers. From the defensive perspective, it may be tempting to call for an authority to forbid and punish all bluffing. This would be costly in small games (i.e., games among ordinary people) and impossible in large games (i.e., games among nations). The small games, at least, can be a source of fun. As to the large games, we just have to hope for the best.