Game Theory and Dominant Strategy in Cosmic Encounter (1977)
Game Theory and Dominant Strategy in Cosmic Encounter (1977)
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Cosmic Encounter is a hobby board game published in 1977 and designed by Peter Olotka, Jack Kittredge, Bill Eberle, and Bill Norton. For more than 40 years, the game has been cherished by casual and hobby gamers alike, and for good reason. The game’s design is intended to force rival players into tense game-theory conflicts. In this post, I will outline a simplified ruleset of the game, and the 2-player payoff matrices it creates over the course of a single play session.
Note: My main citation is the rulebook of Cosmic Encounter, which can be downloaded as a PDF from [1]
In Cosmic Encounter, each player (3-5 players) plays as one Alien race attempting to win enough battles, and conquer enough planets, to win. Each turn, one player will be chosen as the Attacker, and another random player will be chosen as the Defender. These two players have time to discuss their encounter, try to reach an agreement, or simply bluff and intimidate each other. Finally, both players simultaneously play a card from their hand, and these cards are resolved to determine the winner of the battle.
For the purposes of this post, I will only describe the two most common cards – Attack and Negotiate. Attack cards have a number from 5-50, corresponding to the strength of the attack. If both players play attack cards, the higher card wins the battle. Negotiate cards are “concessions.” If both players play a negotiate card, the battle is called off, and the players may freely trade resources. However, if one player plays an Attack card, and the other plays a Negotiate, the Attack card will always beat the negotiate and win the battle.
Because Cosmic Encounter forces players into 2-person battles, we can easily set up a 2×2 payoff matrix for a battle. Each player has the option to play either an Attack or Negotiate. The normalized payoffs are approximately as follows:
https://ibb.co/n3VZLf
The choice with the highest payoff to any player is to attack when their opponent negotiates. No matter the strength of their Attack, the attacking player will win and the opposing player will lose. On average, players have equal payoff when both attacking, since the probability that A has a higher attack card than B is roughly 1/2. The players will also receive equal payout upon a successful negotiation (Neg-Neg) because neither will be incentivized to give up more than they receive in the trade.
However, successfully negotiating with result in greater gains for each player on average, since the battle is cancelled and the players will not suffer any casualties as a result.
There is only one pure strategy equilibrium in this simplified view of Cosmic Encounter – both players attacking one another. If Player A knows that Player B will negotiate, Player A is incentivized to attack. Since player B knows this, they will also attack in order to prevent their own loss in battle. Notice that the Nash equilibrium (Atk-Atk) actually has an alternative which is strictly better for both players (Neg-Neg) but will rarely be used, since one player can make a greedy choice which forces the other player to follow.
This matrix is incredibly structurally similar to a Peace-War Game [3], (as well as thematically similar) in which two countries profit from mutual Peace, but profit even more if they deviate and declare war on a peaceful opponent. Since both players will play defensively to minimize loss, this results in mutual War, an objectively worse choice than mutual Peace, yet still a Nash Equilibrium.
This payoff matrix illustrates why Cosmic Encounter is engaging to players. Each decision is made not based on chance, but on careful consideration of the opponent’s psyche and likely moves. More mechanics, such as negotiation, discussion, allies, and special cards, complicate the game’s decisions even further, and add additional emergent behaviors.
For example, each player of Cosmic Encounter will have one alien power, which changes the rules of the game. One alien power in particular, “Pacifist”, drastically modifies this payoff matrix. If the Pacifist plays a Negotiate card, and their opponent plays an Attack card, the Pacifist will actually win the battle, as if they had played a larger Attack card. A payoff matrix for a battle involving a player with this power might resemble the following:
https://ibb.co/kyVZLf
Such a small rule change, as you can see, drastically changes the way the game is played. Now, the player with “Pacifist” will always profit at least as much as player A, making them a dangerous player to fight. This payoff matrix no longer has a Nash Equilibrium; If both players will Attack, the Pacifist is incentivized to Negotiate. A will then consider Negotiating, but Pacifist will want to Attack in response. Changing one player’s rules has drastically increased the complexity of a battle – no longer is war the most likely outcome, the outcome is now very difficult to predict.
If you are interested in Game Theory or its applications, I would strongly recommend getting together with some friends and playing Cosmic Encounter. Of all the games I have played, it best exemplifies a well-balanced game representative of game-theory concepts, while still being fun and engaging for everyone involved. If you are unable to obtain a physical copy, Cosmic Encounter can be played online using Table Top Simulator. Instructions can be found at [2].
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[1]“Cosmic Encounter.” Fantasy Flight Games, Fantasy Flight Games, 2018, www.fantasyflightgames.com/en/products/cosmic-encounter/.
[2]“Tabletop Simulator.” Cosmic Encounter Connector, HBO, 2017, www.cosmicencounter.com/playnow/.
[3]https://en.wikipedia.org/wiki/Peace_war_game