# Survivor: Mathematical predictions of the big win

Eighteen survivors battled it out for 39 days in Fiji for their chance to claim the million dollar prize in the season finale of Survivor: Heroes vs. Healers vs. Hustlers. The final five survivors — Ben, Chrissy, Devon, Mike and Ryan — competed for a chance at the top spot in the final episode, which aired Dec. 20.

Survivor, now in its 35th season in the United States, is an example of a social game, where social interactions determine the direction of play as well as the winner. As a professor of mathematics, I have studied how the game plays out by applying something known as network science.

If you don’t know how Survivor is played, here is a crash course. Players, called survivors, are placed in a remote location and must provide shelter and food for themselves, with limited support from the outside world. Survivors are split into two or more tribes that live and work together.

Tribes compete for immunity, and the losing tribe goes to tribal council, where one of their members is voted off the game. At some point during the season, the tribes merge, and the remaining survivors continue to compete for individual immunity.

Survivors voted out may become part of the jury. Once there are a small number of survivors left (typically, two or three), the jury votes in favour of one of them to become the Sole Survivor who receives a cash prize of \$1 million.

## Enter the network science study

I conducted a network science study on Survivor with my Master’s student, Rehan Malik, as part of a project on dynamic competition networks. Network science is an interdisciplinary field that studies complex relationships to better understand their structure and predict their development. Networks are composed of a set of “nodes” and the links between them are called “edges.” From our social interactions on Facebook, to websites and their links, to biochemical interactions between proteins in our cells, networks arise everywhere.

Nodes in Survivor represent players, and there is an edge from player A to player B if A votes for B. For example, if Chrissy votes for Mike, then we represent that by a directed arrow from the node representing Chrissy to the node representing Mike. The network evolves with new votes added after each episode.

Whether considering negatively correlated stocks, duelling gangs or competing athletes, competition is a fact of life in real-world networks. While social interaction is often studied from the premise of friendship or other positive social interaction, we take the view that the “edges” in our network correspond to competition or rivalry.

A simple indicator of top players is their “in-degree,” which is the number of votes they have against them. Top players tend to have low in-degree, as is the case with Devon who has an in-degree of zero so far in Survivor: Heroes vs. Healers vs. Hustlers.

Another crucial measure is the “number of common out-neighbours.” If two players — say Ryan and Lauren — vote for Ben, then Ben is their common out-neighbour. A player’s “common out-neighours score” (or “CON score”) is the count of common out-neighbours they have in common with all other players.

We’ve noticed that top players tend to have high CON scores. For example, in the first season of Survivor set in Borneo, Richard had one of the highest CON scores of 29 (surpassed only by Rudy with 33 and Susan with 32, who finished third and fourth, respectively). In Survivor: Heroes vs. Healers vs. Hustlers, Ryan has the highest CON score of 44.

Another key measure of top players is their “closeness.” Nodes in the network may be joined by paths, and the length of a shortest path between two nodes is their “distance.” For the closeness of a node A, we sum up the distances from A to all other nodes, then form the ratio of 1 over this sum.

Intuitively, if a player has high closeness, then they are centrally located in the network. We discovered that closeness emerged as a reliable predictor of top players over several seasons of Survivor.

As an example, in Survivor: Borneo, Sole Survivor Richard had the highest closeness of 0.74 versus other players that season.

## And the winner is

Before the series finale, we looked at how network science could predict the winner of Survivor: Heroes vs. Healers vs. Hustlers? Ryan had the highest closeness of 0.73, followed by Chrissy with 0.7, and Devon with 0.68. Mike and Ben had in-degrees six and eight, respectively, and lower closeness than Devon.

Ryan may have been the favoured finalist because he had the highest closeness, highest CON score, and in-degree two. Devon had in-degree zero, while Chrissy had in-degree seven, so that may have given Devon an advantage among the two.

As a caveat, network science is only one predictive tool for the game, which has twists undetectable to our analysis, such as gaining immunity (which I noted may have given Ben the upper hand this season).

Our analysis also considers “alliances,” which are groups of individuals who pool social capital to go further in the game. Alliances tend to be sets of nodes with “low edge density:” when the ratio of the number of edges between nodes in the set versus the number of pairs of nodes is small.

In the most recent episode before the finale, a new alliance with Chrissy, Devon and Ryan was formed after Chrissy and Devon won a reward challenge. That alliance appeared strong with edge density zero, further bolstering one of the three’s chances to win.

However, the winner of Survivor: Heroes vs. Healers vs. Hustlers turned out to be Ben Driebergen! After playing the surprise Hidden Immunity Idol three times, he secured his position in the final four and after beating Devon in a fire-making challenge, he made it to the final three. In the end, Ben won five jury votes, with Chrissy winning two votes, and Ryan winning one.

Based on our analysis, the most likely final three would have been Chrissy, Devon and Ryan. However, our analysis is insensitive to plays such as the Hidden Immunity Idol. While network science can help narrow down potential winners and alliances, Survivor contains twists and turns that make it anyone’s game.

For those future survivors out there looking for advice, your goal should be to maintain high closeness, low in-degree, have a high CON score, and form low edge density alliances.

That is, make strong alliances, few enemies, and position yourself as a central player. Winning the million dollar prize may depend on it.

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