Cognitive Biases in Strategic Decision-Making

1. Framing Effects: It's All in How You See It

One of the most powerful cognitive biases is the framing effect. This occurs when our decisions are influenced by the way information is presented, even when the underlying facts are identical. A choice framed in terms of potential gains can elicit a very different response than the exact same choice framed in terms of potential losses.

The seminal work explaining this phenomenon is Prospect Theory, developed by psychologists Daniel Kahneman and Amos Tversky, for which Kahneman won the Nobel Prize in Economics.

As you saw, people's preferences flip completely depending on whether the outcomes are framed as "lives saved" (a gain) or "lives lost" (a loss).

• Gains Frame: When choosing between saving 200 people for sure and a 1/3 chance of saving all 600, most people prefer the certain gain. This is risk-averse behavior.
• Losses Frame: When choosing between 400 people dying for sure and a 2/3 chance of all 600 dying, most people prefer the gamble. They take a risk to avoid a certain loss. This is risk-seeking behavior.

This inconsistency violates a basic assumption of rational choice theory. Prospect theory explains why this happens.

The Core Principles of Prospect Theory

To understand framing, we need to understand the psychological model of value that underpins it.

Let's summarize the key components mathematically and graphically. Prospect theory replaces the standard utility function with a value function, $v(x)$, that has three main properties:

1. Reference Dependence: Value is defined over changes in wealth (gains and losses) relative to a reference point $r$, not absolute levels of wealth. The argument of the function is $x-r$. The framing of the Asian Disease problem effectively shifted the reference point from "zero people dead" to "600 people dead".

2. Loss Aversion: The function is steeper for losses than for gains. Formally, for any $x>0$, $v(-x)<-v(x)$. A loss of $100 feels worse than a gain of $100 feels good. This is captured by the "kink" at the origin in the value function.

3. Diminishing Sensitivity: The function is concave for gains ($v^{\prime \prime }(x)<0$ for $x>0$) and convex for losses ($v^{\prime \prime }(x)>0$ for $x<0$). This leads to risk aversion in the domain of gains and risk seeking in the domain of losses.

These features combine to create the distinctive S-shaped value function.

2. Anchoring: The Power of First Impressions

A second major cognitive bias is anchoring. This is our tendency to rely heavily on the first piece of information we receive—the "anchor"—when making subsequent judgments. Even when the anchor is completely irrelevant, it can have a powerful pull on our decisions.

This effect is not just a psychological curiosity; it can be formalized. Just as you are comfortable with physical models defined by parameters, we can model anchoring with a simple, yet powerful, mathematical structure.

Let's crystallize the formal model. When evaluating a lottery $L$, an agent doesn't necessarily calculate its true expected utility, $EU(L)$. Instead, their subjective valuation, $V(L)$, is determined by a process of anchoring and adjustment.

1. Anchor: The agent forms an anchor, $A(L)$, often by simplifying the problem. A common simplification is to ignore probabilities and just average the utility of the outcomes. For a discrete lottery with $n$ outcomes $x_i$, the anchor is $A(L)=\frac{1}{n}{\sum }_{i=1}^{n}U(x_i)$.

2. Adjust: The agent then adjusts from this anchor in the direction of the true expected utility. However, this adjustment is insufficient.

The final valuation is a weighted average of the anchor and the true expected utility:

where $\gamma \in [0,1]$ is the adjustment parameter.

• If $\gamma =1$, the agent adjusts fully and behaves like a standard expected utility maximizer.
• If $\gamma =0$, the agent ignores probabilities entirely and relies only on the anchor.
• For $0<\gamma <1$, the agent is biased towards the anchor.

This elegant model provides a bridge between perfectly rational behavior and heuristic-driven decision-making, controlled by a single parameter, $\gamma$, which you can think of as a measure of the agent's susceptibility to the anchoring bias.

3. Cognitive Biases in Strategic Games

How do these biases affect strategic interactions? They can fundamentally change how players perceive a game and what strategies they choose, leading to outcomes that classical game theory cannot explain.

In our last lesson, we saw how a level-k model begins with a non-strategic L0 player. Cognitive biases like framing and anchoring are excellent candidates for describing the "naïve" behavior of L0 players. More sophisticated L1 and L2 players then best-respond to this biased behavior.

Let's look at an example from the "Behavioral Game Theory" slides by Vincent Crawford, which we used in the previous lesson.

Consider a simple Hide and Seek game described in Section 53 of the Crawford slides (LINK). A Hider can hide a prize in one of four boxes, labeled A, B, A, A. A Seeker can open one box. If the Seeker finds the prize, the Seeker gets 1 point; otherwise, the Hider gets 1 point.

The unique Nash Equilibrium (as in Matching Pennies) is for both players to choose each location with a probability of 1/4. In equilibrium, the labels on the boxes are irrelevant.

However, experimental results show that choices are far from random:

• Hiders overwhelmingly choose the central 'A' (the one between B and another A).
• Seekers also choose the central 'A' most often, even more so than Hiders.

Why? The non-neutral framing of the locations creates salience.

• The 'B' box is unique by its label.
• The two end 'A' boxes are unique by their position.
• This makes the central 'A' the "least salient" or "most normal" location. It feels like the best place to hide something.

A level-k model incorporating this framing can explain the observed behavior:

• L0 (Hider/Seeker): A non-strategic player might be drawn to the most salient locations (B or the ends).
• L1 Hider: Believing the Seeker is L0 and will look in a salient location, the L1 Hider chooses the least salient location: the central 'A'.
• L1 Seeker: Believing the Hider is L0 and has hidden in a salient spot, the L1 Seeker looks in a salient spot.
• L2 Seeker: Believing the Hider is an L1 who has hidden in the central 'A', the L2 Seeker looks in the central 'A'.

The model predicts that a mixture of player types would lead to an over-representation of choices for the central 'A', especially for Seekers (who can be L2), just as the data shows. Here, the framing acts as an anchor for the reasoning process, pulling behavior away from the symmetric equilibrium.

Conclusion

Today, we've added another layer to our understanding of behavioral game theory. While the previous lesson showed that people's cognitive capacity is limited, today we saw that their cognitive process is systematically biased.

Key Takeaways:

• Framing Effects, explained by Prospect Theory, show that choices are not based on absolute outcomes but on gains and losses relative to a reference point. People tend to be risk-averse for gains and risk-seeking for losses.
• Loss Aversion is a critical component, where losses have a greater psychological impact than equivalent gains.
• Anchoring is the tendency to be overly influenced by an initial piece of information, even if it is arbitrary. This heuristic can be formalized with a mathematical model where a player's valuation is a weighted average of an anchor and the true expected utility.
• In strategic games, these biases can define the behavior of "naïve" players, which more sophisticated players must anticipate. They create focal points and asymmetries that can pull outcomes far from the predictions of classical game theory.

Preview of the Next Module:
We have now completed our survey of the foundations of game theory, from classical equilibrium concepts to the behavioral critiques of social preferences and cognitive biases. In the next module, we will begin applying these powerful tools to analyze specific, important real-world strategic environments. Our first topic will be Auctions and Bidding Strategy, a setting where understanding others' valuations—and the biases that shape them—is the key to success.