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Game Theory in Game Shows
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Golden Balls was a British daytime game show which aired on the ITV Network from 2007 to 2009. You can search the intricacies of the game, but after playing some rounds prior to accumulate as much winnings as possible, two contestants are given two sets of balls (“Split” or “Steal”) and are able to converse with one another before making their ultimate decision.1. If both choose “Split”, they receive half the jackpot; 2. If one chooses “Split” and the other “Steal”, the Steal contestant wins the entire jackpot and the Split contestant receives zero; or 3. If both choose “Steal”, they both leave with nothing.
From the above, you might ask yourself “why I have brought this up?” or “what does this have to do with economics?” it turns out there is more to this than meets the eye…
Game Theory is the study of multi-person decision problems. Such problems arise frequently in economics, ranging from oligopolies, as each firm must consider the strategic behaviour of the others, to trading processes (like bargaining and auction models), or to collusion between nations in choosing tariffs and other trade policies.
For now, let us focus on the “Split or Steal” decision described above. In Game Theory, this is a game of the following simple form: the players simultaneously choose actions (this simultaneous-move game is known as a static game); each player’s payoff is common knowledge (known as complete information); and the players receive payoffs that depend on the combination of actions just chosen. Knowing the rules of the game, and assuming we have accumulated a jackpot of $50,000, we can present the potential payoffs in the following payoff matrix:
A payoff matrix, simply, is a depiction of all possible outcomes (or the amount received) when two (or potentially more) groups have to make a strategic decision. What is important to understand is that the payoffs for the ‘row’ player (in this case, Nick) are shown first. Conversely, the payoffs for the ‘column’ player (you, the Reader) are shown second. Note that because each cell has two payoff entries, this is formally referred to as a payoff bi-matrix.
To understand how to read the matrix, it helps to go through the above by each decision. First, let’s assume the Reader decides to choose “Split.” Now, we are going to completely ignore the Steal column,
As mentioned, the payoffs for Nick are the first entries in matrix. If Nick decides to split, he will receive a payoff of $25,000. However, if he decides to steal, he will receive a payoff of $50,000.
Now, I don’t know about you but $50,000 sounds much better than $25,000. So, with this logic, Nick has an incentive: he will play the Steal strategy.
Now, let us assume you decide to pick Steal,
As can be observed from above, the payoffs from either decision are the exact same ($0) and the decision I make is now irrelevant — formally, I am indifferent.
So, what we have is what is called a weakly dominant strategy. This is where the decision (or strategy) we choose provides the same payoff for the other player’s strategy but is strictly greater for one strategy. Alternatively, the Steal strategy is never worse than Splitting, but is sometimes better, as reflected through the payoffs which is either greater or equal.
How about for you? Let’s say Nick decides to split. Ignoring the Steal row, we have the following,
Now we compare the second entries in the payoff matrix. In this case, because $50,000 exceeds $25,000 once more, you have an incentive to choose the strategy Steal.
Conversely, if Nick chooses Steal, ignoring the split row, we are comparing the following,
Again, looking at the second entries, the Reader is indifferent between Split or Steal. Just like Nick, the Reader also has a weakly dominant strategy to steal!
We can present all this as below,
Because each player will now play their weakly dominant strategy to ‘Steal’ we arrive at the following outcome:both players will leave with nothing!
This unique solution is important to understand. The above equilibrium is what is referred to as the Nash equilibrium. This is where each players’ choice is optimal, given the other players decision. Because each player has no incentive to deviate from his or her predicted strategy, and each player’s strategy is the best response to the strategy of the other player, this further solidifies that this prediction is in fact a Nash equilibrium.
The above outcome can also be referred to as Pareto inefficient, as the individuals could do better if they found a way to enforce a co-operative outcome. It turns out that in our game this is possible, as each player is able to negotiate (or bargain) with the other player. If I plea to the other player that I am going to “Split”, things will end badly for me, as it did for this man in the below instance
Remember, if I cry to the other player that I am going to split, rationally, they will choose “Steal”, as their payoff is maximised. But, how about in the following instance:
In this case, Nick was able to manipulate the payoffs of the game and negotiated effectively (or on the appropriate side of the payoff matrix). What we have is the following:
Nick has essentially made a social contract to ensure he is guaranteed to leave with something. When he says, with certainty, he will choose ‘Steal’, Ibrahim’s thought process now changes — he now doesn’t have to consider the ‘Split’ row payoffs. Therefore, Ibrahim’s new dominant strategy (as can be found by comparing the second entries in the bottom row) is to Split. Although, Nick always had the idea to Split regardless, most likely to ensure that he was seen as genuine. Both players are now jumping with glee, leaving £6,800 pounds richer.
But is Nick now seen as fair because he offered a 50/50 split? Not necessarily, as he may have been concerned that if he did offer less, that he may have been turned down and risked leaving with nothing. The notion of fairness and rationality can be extended to another area of economics and a different type of game, but this can be left for another article.
There are plenty of more applications of Game Theory and different strategies that can be discussed. If you would like to see this in a future article/series, feel free to leave a comment!