Notes on combining sequential and simultaneous moves

As part of a series of discussions on Games of Strategy, I've written here about games with sequential moves - those in which players "take turns" and each decision is made with full knowledge of the opponent's last decision - as well as games with simultaneous moves - those in which players make decisions not knowing which option an opponent has selected. Continuing our exploration of game theory, Dr. Wictz and I further discussed games that combine sequential and simultaneous moves.

Notes on simultaneous-move games with continuous strategies

My recent posts on game theory have focused on games with discrete strategies, which is to say that players are faced with a finite number of choices for each decision. Some time ago, in our colloquium on games of strategy, Aaron Honsowetz, Austin Smokowicz, and I discussed games with continuous strategies (also on video) - those games in which players choose a value on a spectrum, such as a price to set on a commodity. 

Notes on games with mixed strategies

In an earlier post regarding theory of simultaneous move games, I concluded with an example of a game between two tennis players that did not demonstrate a Nash equilibrium between its two pure strategies. Sam Hillier: Consulting Philosopher more recently elaborated on the topic with an excellent post on mixed strategies. Whereas I had approached the question of an equilibrium for a single tennis shot and concluded that none existed, a tennis match of course includes many shots, so players have an opportunity to invoke a weighted mix of shots and defenses between the two options.

Notes on simultaneous-move games, and an exploration of the Stag Hunt

Some time ago, the design team of Dr. Wictz and I started discussing the book Games of Strategy by Dixit, Skeath, and Riley.  I wrote on a couple of topics:

• Games of Strategy (an overview of terminology) and
• Games with Sequential Moves

In this post, I'd like to address simultaneous-move games with a specific focus on pure discrete strategies. (We recorded our discussion on this topic in April of last year.)  I recall such games represented in my earliest readings on game theory in the form of a decision-payoff matrix.  In a two-player game in which each player makes a single decision from among a finite number of choices, without knowledge of the other player's decision, the decision-payoff matrix labels the rows with one player's options and the columns with the other player's options.  The corresponding cell for a given combination of decisions yields the payoff to both players. 

Bidding and game theory

I have been thinking about the game theory construct for the share auction in Chicago Express (designer Harry Wu, publisher Queen). The question isn't only one of absolute valuation but also one of the interactive decision-making in the auction. That thought led to consideration of the auction as a game-theory problem.

Notes on Games with Sequential Moves

In this second post in a series exploring games of strategy (begun last month), designers Aaron Honsowetz, Austin Smokowicz, and I explore strategic games involving sequential moves, i.e. those in which each player's decision happens in the context of knowing opponents' previous decisions.  This exploration has its foundation in Chapter 3 of Dixit, Skeath, and Reiley's Games of Strategy.

Notes on Games of Strategy

Over three years ago, I wrote about my effort to approach a simple three-player race game using game theory.  Economist and game designer Dr. Aaron Honsowetz responded, which led to his recommendation that I look up the book Games of Strategy by Avinash Dixit, Susan Skeath, and David Reiley.  I finally obtained the third edition recently, and that has led Aaron, fellow designer Austin Smokowicz, and me to explore Dixit Skeath and Reiley's text in a kind of virtual book club.

Game Theory: A simple multi-player case

Earlier this week I was listening to Episode 36 of the Flip the Table podcast, which discussed the obscure 1979 Bruce Jenner Decathlon Game (publisher Parker Brothers).  The game consists of ten mini-games using an eclectic variety of mechanics.  One of them caught my attention as an elegant bluffing and second-guessing procedure used to resolve the "foot races" in the decathlon.

Revisit: Incan Gold and game theory

[I've been on business travel this week, so in the absence of original material, I'm reposting an article from last spring when I was first discovering Incan Gold.]


We had a family session of Incan Gold this afternoon [original post 16 April 2011].  An interesting development came up when my wife Kathy and I had bailed out of an expedition, and only my two sons Liam and Corey remained to explore the ruins.  One instance each of three different monsters had been turned up, which meant that there was a very real possibility that a second monster of one type would appear and scare the remainder of the party out of the ruins at any point.  But then an artifact showed up, and a very interesting stand-off ensued.  By the rules of the game, if there are two or more people in the expedition, neither gets the artifact, and it stays on the card.  In a subsequent turn, if exactly one of the remaining two people decides to return to his tent, he gets all treasure left on cards from previous turns - including the coveted artifact.  If both players turn back, neither gets the artifact, and the round is over.  If both continue on, both continue to share discovered treasure but risk encountering a monster and losing everything.

What followed was an almost comical staring contest between the two of them to try to figure out whether the other was going to stay or return, and therefore whether to return (in hopes that the other was staying, which would leave the artifact to the returning player) or stay (and keep any subsequent treasure for oneself).

The decision to turn back or to continue is simultaneous among remaining players, so the result is a fairly classic game theory problem, in which the outcome of a decision depends upon an opponent's simultaneous unknown decision.

Own decision  Opponent decides to stay  Opponent decides to go
Stay          Turn over another card    Opponent gets artifact
Go                  Get artifact          Nobody gets artifact


Since "Turn over another card" is mutually risky or mutually beneficial but in no case advantageous for one player over the other if both players stay, then game theory would conclude that the only logical decision would be to go.  But if both players decide to go, then neither gets the artifact.

The piece that's missing in my decision table above, however, is that if either player stays, another card will be turned over, to the risk or benefit of the player(s) staying.  So there might be an advantage to staying if a player perceives a potential treasure greater than getting the artifact.  But that's really unlikely, in fact, so the stand-off will typically end up in both players going back and neither getting the artifact. Having said that, however, the game actually plays unpredictably, and perceived risk and reward tend to rule over cold logic.

We've really come to like this risk management game.  I'm apparently way too conservative, however.  I came in last today, and Corey (10) beat us all.  (I seem to recall that he ended up with the artifact more than once, by the way.)

Incan Gold and Game Theory

We had a family session of Incan Gold (or, more precisely, my home-made knock-off) this afternoon.  An interesting development came up when my wife Kathy and I had bailed out of an expedition, and only my two sons Liam and Corey remained to explore the ruins.  One instance each of three different monsters had been turned up, which meant that there was a very real possibility that a second monster of one type would appear and scare the remainder of the party out of the ruins at any point.  But then an artifact showed up, and a very interesting stand-off ensued.  By the rules of the game, if there are two or more people in the expedition, neither gets the artifact, and it stays on the card.  In a subsequent turn, if exactly one of the remaining two people decides to return to his tent, he gets all treasure left on cards from previous turns - including the coveted artifact.  If both players turn back, neither gets the artifact, and the round is over.  If both continue on, both continue to share discovered treasure but risk encountering a monster and losing everything.

What followed was an almost comical staring contest between the two of them to try to figure out whether the other was going to stay or return, and therefore whether to return (in hopes that the other was staying, which would leave the artifact to the returning player) or stay (and keep any subsequent treasure for oneself).

The decision to turn back or to continue is simultaneous among remaining players, so the result is a fairly classic game theory problem, in which the outcome of a decision depends upon an opponent's simultaneous unknown decision.

Own decision  Opponent decides to stay  Opponent decides to go
Stay          Turn over another card    Opponent gets artifact
Go                  Get artifact          Nobody gets artifact


Since "Turn over another card" is mutually risky or mutually beneficial but in no case advantageous for one player over the other if both players stay, then game theory would conclude that the only logical decision would be to go.  But if both players decide to go, then neither gets the artifact.

The piece that's missing in my decision table above, however, is that if either player stays, another card will be turned over, to the risk or benefit of the player(s) staying.  So there might be an advantage to staying if a player perceives a potential treasure greater than getting the artifact.  But that's really unlikely, in fact, so the stand-off will typically end up in both players going back and neither getting the artifact. Having said that, however, the game actually plays unpredictably, and perceived risk and reward tend to rule over cold logic.

We've really come to like this risk management game.  I'm apparently way too conservative, however.  I came in last today, and Corey (10) beat us all.  (I seem to recall that he ended up with the artifact more than once, by the way.)