Golden Balls

A game where cheap talk is essential.

Introduction

In the British television show Golden Balls two contestants play at the end of each episode the game Split or Steal?. The game is a variant on the classic prisoner's dilemma: each contestant chooses independently to split the jackpot or to steal the jackpot, and their payoff will be decided based on the combination of their own choice and the choice made by the other contestant. Mostly, the stakes are large, between £ 3 and £ 100,150 with an average of £ 13,300. So, Golden Balls is not just a game.

Like for the prisoner's dilemma, there is a payoff matrix for Split or Steal?, shown in the following table. The difference with the prisoner's dilemma is that the payoff is different in some way. The payoffs for the four possible outcomes are displayed in table below. If we assume that each player only cares about maximising her immediate financial payoff, the question Split or Steal? can be seen as a weak form of the prisoner's dilemma. In the prisoner's dilemma, defecting is a dominant strategy for both players, but for Split or Steal? it is not. Ortmann found that the cooperation rate is higher in a form of the prisoner's dilemma with weakly dominant strategies than in one with strictly dominant strategy.

There is also one other component in Split or Steal? that we need to mention. The round starts with a limited time where both players can talk about their imminent decision between 'split' or 'steal'. This time can be used to convince the other player about one's choice and what the other player should choose. What is the influence of communication on the final decision? Can you change the rules of the game by communication?

Influences on the final choice

Two studies analysed the behavior of the contestants of Golden Balls. Van den Assem found that players do not seem to be more likely to cooperate if their opponent might be expected to cooperate. Darai found that handshakes decrease the likelihood to cooperate, but if players mutually promise each other to cooperate and in addition shake hands on it, the cooperation rate increase. Both studies found that males are less cooperative than females, but men become increasingly cooperative as their age increase.

The game is also discussed an analyzed in many blogs. For example Bruce Schneier, discussed the strategy of Nick. Nick does not trust his opponent and therefore he changes the game. He offers to split the pot outside the game – set up a meta-game of sorts – and removes his opponent's incentive to lie.

Schneier states that Nick changes the variant of the Prisoner's Dilemma game into a classic Trust game:

"Player 1 gets a pot of money and gives some percentage of it to player 2. The money is then multiplied by some amount and player 2 gives some percentage of it back to player 1. In a classic rational self-interest model, it makes no sense for player 2 to give any of the money back to player 1. Given that, it makes no sense for player 1 to give any of the money to player 2 in the first place. However, if player 1 gives player 2 100%, and player 2 gives player 1 back 50% of the increased pot, they both end up the happiest."

"Nick sets himself up as player 2, promising to give his opponent 50% of the jackpot outside of the game. The opponent is now player 1, deciding whether to give Nick the money in the first place. There is one different with the traditional trust game: the opponent cannot keep the money if he doesn't give it to Nick. So he might as well give the money to Nick. The game is turned on its head; trusting Nick now means letting him have all the money. Not trusting Nick means… well, it doesn't mean anything anymore."

Cheap talk and signaling

The studies discussed above found some influences on the final choice; this reinforces the idea that the final choice can be influenced since there is just a weakly dominant strategy. However, the signals that the contestants gave each other in the show could not directly affect the final payoff. A player can say that he will choose split, but he can lie. Therefore we call the communication in the show 'cheap talk'.

Farell showed that cheap talk does not ensure that players will play a Nash equilibrium or that an efficient outcome will emerge or that all information will be shared. However, cheap talk will affect the listener's belief. Then, the question is: How could we model the beliefs of the players?

Modelling the game

For this project we study how to model the influence of communication in Split or Steal? on the final result. There are many different models possible, so we first need to decide what a model needs to show. Therefore we asked ourselves the question: for what purpose would we use to model? We came to the conclusion that we would like to search for a model that could represent the actual situation during the Split or Steal? game the best, so it could help future players to make decisions in a particular situation.

First it is important to mention some important components that need to be incorporated in the model:

Communication between two agents,
Beliefs of the agents,
Change of information and/or change of beliefs.

Next to these main components it is important that the model could deal with lies and trust or distrust. Also it is important that the model is able to represent 'new' information or new rules. We saw in the example of player Nick that it is possible to change the rules of the game a little bit.

First, we choose to model the possible Kripke worlds that could be the case when an agent is speaking the truth or is a liar. A first analysis of the change of Kripke worlds in this game is given in the section Knowledge Models. We also made an implementation to demonstrate the possible Kripke models in different situation.

Then we would like to use a BDI-approach. We think that a BDI-approach could better model the actual situation, since it makes a distinction between beliefs, desires and intentions of the players at different time steps. It gives not only insight in the possible situations, but also an insight in the 'bigger plan' of a contestant.

A third approach is to use a combination of dynamic epistemic logic and game theory as formulated by Gerbrandy. An advantage of this method is that it has a dynamic component and is therefore able to show how information changes in a multi-agent setting.

Knowledge models

First of all, we define p_i and q_i for i ∈ { 1, 2 } as follows.

p_i : player i chooses the split ball
q_i : player i chooses the steal ball

Furthermore, observe that, since a player can choose exactly one ball, we have for all i ∈ { 1, 2 }:

p_i ⇔ ¬q_i

We will model the situation that player 2 (Nick) says he will choose the steal ball, and if player 1 will choose the split ball, player 2 will split the money after the show.

The payoff for the players changes.

Besides that, he actually makes the left column, in which player 2 chooses split, not an option. Assuming that everything he says is true, we obtain the following Kripke model.

However, he might lie about the fact that he chooses the steal ball. In this case it does not matter whether he would split the money after the show, since he will not obtain all of the money in any case and we obtain the same model either way.

Furthermore, he also might lie about the fact that he would split the money after the show, in this case we obtain the following model.

BDI

In Rao and Georgeff's BDI approach, an agent has a belief, a desire and an intention that can change in time. Rao en Georgeff's model is a branching-time possible-worlds model that was especially made to formalize intention. Two of the main axioms in this BDI-approach are the

Axiom of belief-goal compatibility and the
Axiom of goal-intention compatibility.

According to the first axiom, it is only possible that an agent considers a goal φ to be true, when there is at least one path in all the belief-accessible worlds where this goal φ is true. The second axiom states that there is no intention φ possible, if there is no path in all the goal-accessible worlds where this intention φ is true.

We modelled the strategy of player Nick and his opponent as well in R&G's BDI-model. An advantage of modelling the game according to this BDI-approach is that is easy to visualize the goals and intentions of the agent and the paths the need to follow to achieve their intention and so their goal. Since their goals and intentions are always dependent on their beliefs, this model gives a good insight in how an agent could think out a strategy given a certain belief.

A disadvantage of using this BDI-approach is that it cannot show the communication among the agents and how announcements change the beliefs of the agents.

Conclusion

First of all, we showed how displaying possible Kripke worlds could give players of the Split or Steal? game insight in the game and implemented a tool that a user could use to make the best decision based on the belief they have. Secondly, we showed how R&G's BDI-approach can illustrate the plans that players could have and how they can achieve a goal.

Both methods use beliefs as 'given'. However, having the right beliefs is one of the most crucial things in Split or Steal?. Can you trust your opponent? That is the big question. As we saw in the introduction, statistics of the behaviour of contestants could possibly help you in making a decision, but would you really make a choice on gender, age or a handshake?

Credibility

When is an announcement credible? Gerbrandy formulated credibility as:"a public announcement is credible by agent A is credible to agent B in a game G iff it holds for all games G’ that B considers possible in G that the public announcement of φ has a positive value for A in G’ iff φ is true in G’".

Gerbrandy used this concept of credibility to make models that are a combination of dynamic epistemic logic and game theory. An advantage of this method is that it has a dynamic component and is therefore able to show how information changes in a multi-agent setting. In the example of player Nick, we saw that the opponent could trust the announcement that Nick always would choose steal. The opponent knew that Nick did not trust him and therefore we can call Nick's announcement credible. However, in most of the situations where two player play the split or steal game, credibility would not be useful.

An approach similar to Gerbrandy's was presented by Van Ditmarsch et al. The approach of van Ditmarsch makes use of doxastic logic and dynamic logic to describe the effect of lying in a game. Van Ditmarsch et al. describe lying as agent A believing ¬p, while he communicates p. We think that this approach would be very useful while analysing our example, especially when it is extended with probabilities.

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